On the abundance of $k$ -fold semi-monotone minimal sets in bimodal circle maps

Philip Boyland Department of Mathematics
University of Florida
372 Little Hall
Gainesville
FL 32611-8105, USA [email protected]

Abstract.

Inspired by a twist maps theorem of Mather we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map $g$ to the $k$ -fold cover. For each irrational in the interior of the rotation set the collection of the $k$ -fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$ -dimensional ball in the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool is a generalization of a construction of Hedlund and Morse which generates the symbolic analogs of these $k$ -fold well-ordered invariant sets.

1. introduction

In a dynamical system a rotation number or vector measures the asymptotic speed and direction of an orbit. The rotation set collects all these together into a single invariant of the system. The natural question is how much does this invariant tell you about the dynamics? Perhaps the first issue is whether for each rotation number is there a nice invariant set in which every point has that rotation number?

This question has been studied in a number of contexts with the most complete answer known about maps of the circle, annulus and two-dimensional torus. In these cases the basic question is enhanced by requiring that the invariant set of a given rotation vector has the same combinatorics as a rigid rotation. So, for example, for a continuous degree-one map $g$ of the circle and a number $\omega$ in its rotation set is there an invariant set $Z_{\omega}$ on which the action of $g$ looks like the invariant set of rigid rotation of the circle by $\omega$ ? This is made more precise and clear by lifting the dynamics to the universal cover ${\mathbb{R}}$ . The question then translates to whether the action of the lift $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ on the lift ${\tilde{Z}}_{\omega}$ is order-preserving? For this class of maps the answer is yes; such invariant sets always exist ([17]).

On the torus and annulus a general homeomorphism isotopic to the identity lacks the structure to force the desired invariant sets to be order preserving so topological analogs are used ([30, 9, 40]). The required additional structure available in the annulus case is the monotone twist hypothesis. In this case the celebrated Aubry-Mather Theorem states that for each rational in the rotation set there is a periodic orbit and for each irrational a Denjoy minimal set and the action of the map on these invariant sets is ordered in the circle factor like rigid rotation. These invariant sets are now called Aubry-Mather sets.

For an area-preserving monotone twist map the minimal set with a given irrational rotation number could be an invariant circle. When a parameter is altered and this circle breaks it is replaced by an invariant Denjoy minimal set. In [35] Mather investigated what additional dynamics this forces. He showed that in the absence of an invariant circle with a given irrational rotation number there are many other invariant minimal Cantor sets with the same rotation number and the dynamics on these sets is nicely ordered under the dynamics not in the base, but rather in finite covering spaces of the annulus.

More specifically, these invariant minimal sets are Denjoy minimal sets which are uniquely ergodic. Their collection is topologized using the weak topology on these measures. Mather showed that for a given irrational rotation number in the rotation set the collection of Denjoy minimal sets with that rotation number which are ordered in the $k$ -fold cover contains a topological disk of dimension $k-1$ . In this paper we prove the analog of this result for a class ${\mathcal{G}}$ of bimodal degree-one maps of the circle as well as describing their periodic orbits which have nicely ordered lifts in the $k$ -fold cover.

Mather’s proof use variational methods. The main methods here come from symbolic dynamics and utilize a construction that is a generalization of one due to Hedlund and Morse ([37] and [23] page 111). Such generalizations are a common tool in topological dynamics ([34], [3] pages 234-241, and [10]). This HM construction used here for a rotation number $\omega$ and number $k$ generates the itineraries under rigid rotation by $\omega$ with respect to an address systems made from $2k$ intervals on the circle. The closure of this set of itineraries yields the symbolic analog of an invariant set that is nicely ordered in the $k$ -fold cover. These sets are termed symbolic $k$ -fold semi-monotone sets (symbolic kfsm sets). Varying the address system parameterizes the symbolic kfsm sets in both the Hausdorff and weak topologies.

A physical kfsm set (or just a kfsm set if the context is clear) is a $g$ -invariant set $Z$ which has a lift $Z^{\prime}$ to the $k$ -fold cover of the circle $S_{k}$ on which the lift $g_{k}$ of $g$ acts like rigid rotation. Physical and symbolic kfsm sets are connected by a second main tool.

The second tool again uses addresses and itineraries, but this time to code orbits under the bimodal map $g$ . Restricting to all orbits that land in the positive slope region we get an invariant set $\Lambda(g)$ which is coded by an order interval in the one-sided two shift $\Sigma_{2}^{+}$ . Since we need to study invariant sets that are ordered in the $k$ -fold cover we lift this coding to one on the orbits which stay in the positive slope region under $g_{k}$ in the $k$ -fold cover $S_{k}$ . This yields a $g_{k}$ -invariant set $\Lambda_{k}(g)$ which is then coded by a subshift ${\hat{\Lambda}}_{k}(g)\subset\Sigma_{2k}^{+}$ .

This result connects the physical kfsm sets in $\Lambda_{k}(g)$ , the symbolic kfsm sets in ${\hat{\Lambda}}_{k}(g)$ , and the symbolic sets generated by the HM construction. Part (c) will be explained below.

Theorem 1.1.

For $g\in{\mathcal{G}}$ the following are equivalent:

(a)

$Z\subset\Lambda(g)$ is a recurrent kfsm set for $g$ .
(b)

The symbolic coding of $Z$ via the itinerary map is constructable via the HM process.
(c)

$Z$ is a recurrent set of an interpolated semi-monotone map $H_{\vec{c}}$ in the $k$ -fold cover.

Note that the result is restricted to recurrent kfsm sets. There are several reasons for this. First, recurrence is where the interesting dynamics occurs, second, invariant measures are always supported on recurrent sets, and finally, the HM construction produces recurrent sets. As is well-known in Aubry-Mather theory there are also nonrecurrent kfsm sets which consist of a recurrent set and orbits homoclinic to that set. We also restrict to orbits that stay in the positive slope region of $g$ . Considering kfsm sets that also have points in the negative slope region at most adds additional homoclinic orbits or shadow periodic orbits. See Section 13.2.

For each $k$ , the HM construction depends on two parameters, a rotation number $\omega$ and a parameter $\vec{\nu}$ describing the address system on the circle. For a rational rotation number it produces a finite cluster of periodic orbits while for irrationals it produces a semi-Denjoy minimal set. Since $g$ is noninjective the analogs of Denjoy minimal sets have pairs of points that collapse in forward time, and hence the “‘semi” in their name.

Another main result is that the HM construction parameters $(\omega,\vec{\nu})$ yield a homeomorphic parameterization of the space of invariant measures on the recurrent symbolic kfsm sets with the weak topology. Via the itinerary map, this is pulled back to a parameterization of the space of invariant measures on the physical recurrent kfsm sets with the weak topology. It yields the following result in which $\rho(g)$ is the rotation interval of $g\in{\mathcal{G}}$ .

Theorem 1.2.

Assume $g\in{\mathcal{G}}$ , $\alpha\not\in{\mathbb{Q}}$ , $\alpha\in\operatorname{Int}(\rho(g))$ , and $k>0$ .

(a)

In the weak topology there is $(k-1)$ -dimensional disk of kfsm semi-Denjoy minimal sets with rotation number $\alpha$ .
(b)

If $p_{n}/q_{n}$ is a sequence of rationals in lowest terms with $p_{n}/q_{n}\rightarrow\alpha$ , then the number of distinct kfsm periodic orbits of $g$ with rotation number $p_{n}/q_{n}$ grows like $q_{n}^{k-1}$ .

Informally, a kfsm semi-Denjoy minimal set wraps $k$ -times around the circle with orbits moving at different average speeds in each loop. Lifting to the $k$ -fold cover these “speeds” are given by the amount of the unique invariant measure present in a fundamental domain of $S_{k}$ : more measure means slower speed. The $k$ -dimensional vector of these measures is called the skewness of the minimal set. The sum of the components of the skewness is one and thus the collection of possible skewnesses contains a $(k-1)$ -dimensional ball. The skewness turns out to be an injective parameterization of the kfsm sets for a given irrational rotation number in the interior of the rotation set of a $g\in{\mathcal{G}}$ (see Remark 9.14).

The HM-parametrization of kfsm sets with the Hausdorff topology is only lower semicontinuous. The points of discontinuity are given in Theorem 9.5.

We will on occasion use results derived from those of Aubry-Mather theory. While the context here is a bit different the proofs are virtually identical and so are omitted. There are excellent expositions of Aubry-Mather theory; see, for example, [32], [29] Chapter 13, and [22] Chapter 2. In the context of the generalization of Aubry-Mather theory to monotone recursion maps a version of Mather’s theorem on Denjoy minimal sets is given in [44].

We restrict attention here to a particular class of bimodal circle maps defined in Section 4.1. Using the Parry-Milnor-Thurston Theorem for degree-one circle maps the results can be transferred (with appropriate alterations) to general bimodal circle maps (see Remark 4.2).

It is worth noting that the results here immediately apply to a class of annulus homeomorphisms. This application can be done either via the Brown-Barge-Martin method using the inverse limit of $g\in{\mathcal{G}}$ ([4, 12]) or via the symbolic dynamics in annulus maps with good coding like a rotary horseshoe, for example, [24, 31, 10, 19].

Refer to caption — Figure 1. The lift of a $g\in{\mathcal{G}}$ to the 3-fold cover and an interpolated semi-monotone map

Figure 1 illustrates the conceptual framework that inspired the results here. It shows the graph of a map $g\in{\mathcal{G}}$ lifted to the $3$ -fold cover. At three heights $(c_{1},c_{2},c_{3})=\vec{c}$ the graph is cut-off yielding a semi-monotone circle map $H_{\vec{c}}$ . Such maps have a unique rotation number and well understood recurrent sets which are of necessity semi-monotone sets. As $\vec{c}$ is varied, the rotation number $\rho(H_{\vec{c}})=\rho(\vec{c})$ varies continuously. Thus one would expect that the level sets $\rho^{-1}(\omega)$ provide a parameterization of the kfsm sets with rotation number $\omega$ . In particular, for irrational $\omega$ this level set should be a $(k-1)$ -dimensional disk as in Theorem 1.2(a). This is true for $g\in{\mathcal{G}}$ . Figure 5 below shows some level sets. It is worth noting that this figure is not a bifurcation diagram, but rather a detailed analysis of the dynamics present in a single map.

While providing a valuable heuristic this point of view is not as technically tractable as the HM construction and we content ourselves with just a few comments on it in Section 13. One of which is the addition of item (c) to the list of equivalent conditions in Theorem 1.1.

The literature on bimodal circle map dynamics is vast and we briefly mention only two threads here. Symbolic dynamics for degree one bimodal goes back at least to [5, 26, 25]. The interpolated “flat spot” map trick for finding one-fold semi-monotone sets was discovered and used by many people in the early 80’s; references include [17, 36, 43, 42, 41, 28, 13, 39]. The author learned the trick from G.R. Hall in Spring, 1983 and the idea of applying it in finite covers emerged in conversations with him.

There are many questions raised by this work, but we mention just three here. As is well-known, the one-fold symbolic semi-monotone sets generated by the HM construction are the much-studied Sturmian sequences. The general symbolic kfsm are thus a generalization of one property of the Sturmians to more symbols (there are many other generalizations). The Sturmians have many marvelous properties such as their connection to the Farey tree and substitutions: which properties are shared by symbolic kfsm sets?

The HM construction is an explicit parameterized way of getting well controlled orbits that do not preserve the cyclic order in the base and thus in most cases force positive entropy and as well as other orbits. A second question is how does the parameterization given by the HM construction interact with the forcing orders on orbits in dimension one and two (see, for example, [1] and [11]).

A final question relates to the global parameterization of kfsm by the HM construction. Each bimodal map $\in{\mathcal{G}}$ corresponds to a specific set of parameters, namely, those that generate symbolic kfsm whose physical counterparts are present in $g$ . What is the shape of this set of parameters?

After this work was completed the author became aware of the considerable literature studying invariant sets in the circle that are invariant and nicely ordered under the action of $z\mapsto z^{d}$ [16, 21, 20, 8, 7, 33]. While the exact relationship of that theory and what is contained in the paper is not clear, it is clear that the two areas share many basic ideas and methods. These include using a family of interpolated semi-monotone circle maps with flat spots, tight and loose gaps in invariant Cantor sets, parametrizing the sets using the position of the flat spots, and parametrizing the sets with irrational rotation number by an analog of skewness. Section 14 contains a few more comments on the relationship of the problems.

2. Preliminaries

2.1. Dynamics

Throughout this section $X$ is a metric space and $g:X\rightarrow X$ is a continuous, onto map. Since the maps $g$ we will be considering will usually not be injective, we will be just considering forward orbits, so $o(x,g)=\{x,g(x),g^{2}(x),\dots\}$ .

A point $x$ is recurrent if there exists a sequence $n_{i}\rightarrow\infty$ with $g^{n_{i}}(x)\rightarrow x$ . A $g$ -invariant set $Z$ is called recurrent if every point $z\in Z$ is recurrent. Note that a recurrent subset is usually different than the recurrent set, with the latter being the closure of all recurrent points. A compact invariant set $Z$ is called minimal if every point $z\in Z$ has a forward orbit that is dense in $Z$ .

The one-sided shift space on $n$ symbols is $\Sigma^{+}_{n}=\{0,\dots,n-1\}^{\mathbb{N}}$ and that on ${\mathbb{Z}}$ symbols is $\Sigma^{+}_{\mathbb{Z}}={\mathbb{Z}}^{\mathbb{N}}$ . Occasionally we will write $\Sigma_{\infty}^{+}$ for $\Sigma^{+}_{\mathbb{Z}}$ . For clarity we note that in this paper $0\in{\mathbb{N}}$ . In every case we give one-sided shift spaces the lexicographic order and the left shift map is denoted $\sigma$ , perhaps with a subscript to indicate the shift space upon which it acts. Maps between shifts and subshifts here will always be defined by their action on individual symbols so, for example, $\varphi:\Sigma_{n}^{+}\rightarrow\Sigma_{n}^{+}$ defined on symbols by $s\mapsto\varphi(s)$ means that $\varphi(s_{0}s_{1}s_{2}\dots)=\varphi(s_{0})\varphi(s_{1})\varphi(s_{2})\dots$ . For a block $B=b_{0}\dots b_{N-1}$ in $\Sigma^{+}_{n}$ its cylinder set is $[B]=\{{\underline{s}}\in\Sigma_{n}^{+}\colon s_{i}=b_{i},i=0,\dots,N-1\}$ . Note that all our cylinder sets start with index $0$

The space-map pairs $(X,f)$ and $(Y,g)$ are said to be topologically conjugate by $h$ if $h$ is a homeomorphism from $X$ onto $Y$ and $hf=gh$ .

We will frequently use the standard dynamical tool of addresses and itineraries. Assume $X=X_{0}\sqcup X_{1}\sqcup\dots\sqcup X_{n-1}$ with $\sqcup$ denoting disjoint union. Define the address map $A$ as $A(x)=j$ when $x\in X_{j}$ and the itinerary map $\iota:X\rightarrow\Sigma_{n}^{+}$ by $\iota(x)_{i}=A(g^{i}(x))$ . It is immediate that $\sigma\circ\iota=\iota\circ g$ . In many cases here, $\iota$ will be continuous and injective yielding a topological conjugacy from $(X,g)$ to a subset of $(\Sigma_{n}^{+},\sigma)$ .

We will also encounter the situation where the $X_{j}$ are not disjoint, but intersect only in their frontiers $\operatorname{Fr}(X_{j})$ . In this case we define a “good set” $G=\{x\colon o(x,g)\cap(\cup\operatorname{Fr}(X_{j}))=\emptyset\}$ . In this case the itinerary map is defined as $\iota:G\rightarrow\Sigma_{n}^{+}$ .

For $Z\subset X$ , its interior, closure and frontier are denoted by $\operatorname{Int}(Z),\operatorname{Cl}(Z),$ and $\operatorname{Fr}(X)$ , respectively. The $\epsilon$ -ball about $x$ is $N_{\epsilon}(x)$ . The Hausdorff distance between two sets is denoted $\operatorname{HD}(X,Y)$ . For an interval $I$ in ${\mathbb{R}}$ , $|I|$ denotes it length, and for a finite set $Z$ , $\#Z$ is its cardinality. On an ordered $k$ -tuple the map $\tau$ is the left cyclic shift, so $\tau(a_{1},a_{2},\dots,a_{k})=(a_{2},a_{3},\dots,a_{k},a_{1})$ . On the circle $S^{1}$ , $\theta_{1}<\theta_{2}$ is defined as long as $|\theta_{1}-\theta_{2}|<1/2$ .

2.2. The circle, finite covers and degree one circle maps

While the only compact manifold here will be a circle it will clarify matters to use the language of covering spaces.

In general, if $\pi:{\tilde{Y}}\rightarrow Y$ is a covering space, and $Z\subset Y$ , a lift of $Z$ is any set $Z^{\prime}\subset{\tilde{Y}}$ with $\pi(Z^{\prime})=Z$ . The full lift of $Z$ is ${\tilde{Z}}=\pi^{-1}(Z)$ . If $g:Y\rightarrow Y$ lifts to $\tilde{g}:{\tilde{Y}}\rightarrow{\tilde{Y}}$ and $Z\subset Y$ is $g$ -invariant then the full lift ${\tilde{Z}}$ is $\tilde{g}$ -invariant, a property that is usually not shared by a lift $Z^{\prime}$ .

The universal cover of the circle is ${\mathbb{R}}$ with deck transformation $T(x)=x+1$ and the covering space is thus $\pi:{\mathbb{R}}\rightarrow S^{1}={\mathbb{R}}/T={\mathbb{R}}/{\mathbb{Z}}=[0,1]/\mkern-8.0mu\sim$ . We will only study maps $g:S^{1}\rightarrow S^{1}$ whose lifts $\tilde{g}$ commute with the deck transformation, $\tilde{g}T=T\tilde{g}$ , or $\tilde{g}(x+1)=\tilde{g}(x)+1$ . These circle maps are commonly termed degree one. Our given $g$ will usually have a preferred lift $\tilde{g}$ and so all other lifts are obtained as $T^{n}\tilde{g}$ or $\tilde{g}+n$ .

Central to our study are the finite $k$ -fold covers of the circle for each $k>0$ , $S_{k}={\mathbb{R}}/T^{k}={\mathbb{R}}/k{\mathbb{Z}}=[0,k]/\mkern-8.0mu\sim$ . The deck transformation $T_{k}:S_{k}\rightarrow S_{k}$ is induced by $T$ on ${\mathbb{R}}$ and the covering space is $\pi_{k}:S_{k}\rightarrow S^{1}$ . A preferred lift $\tilde{g}$ of $g$ to ${\mathbb{R}}$ induces a preferred lift $\tilde{g}_{k}:S_{k}\rightarrow S_{k}$ that commutes with $T_{k}$ . We also need the map from the universal cover to the $k$ -fold cover treating $S_{k}$ as the base space $p_{k}:{\mathbb{R}}\rightarrow S_{k}$ .

A $g$ -periodic point $x$ is said to have rotation type $(p,q)$ with respect to the preferred lift $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ if $x$ has period $q$ and for some lift $x^{\prime}\in{\mathbb{R}}$ , $\tilde{g}^{q}(x^{\prime})=T^{p}x^{\prime}$ . Note that there is no requirement here that $p$ and $q$ are relatively prime.

A central concern in this paper is how $g$ -minimal sets in $S^{1}$ lift to $k$ -fold covers.

Theorem 2.1.

Let $g:S^{1}\rightarrow S^{1}$ be degree one and fix $1<k<\infty$ .

(a)

If $Z\subset S^{1}$ is a minimal set, then there exists an $m$ which divides $k$ so that the full lift of $Z$ to $S_{k}$ satisfies

${\tilde{Z}}=\sqcup_{j=1}^{m}Z_{j}^{\prime}$ (2.1)

with each $Z_{j}^{\prime}$ minimal under $\tilde{g}_{k}$ , $\pi_{k}({\tilde{Z}}_{j}^{\prime})=Z$ and $T_{k}(Z_{j}^{\prime})=Z_{j+1}^{\prime}$ with indices $\mkern-8.0mu\mod k$ .
(b)

If $Z^{\prime},Z^{\prime\prime}\subset S_{k}$ are $\tilde{g}_{k}$ minimal sets, we have $\pi_{k}(Z^{\prime})=\pi_{k}(Z^{\prime\prime})$ if and only if $T^{p}_{k}(Z^{\prime})=Z^{\prime\prime}$ for some $p$ .
(c)

If $x\in S^{1}$ is a periodic point with rotation type $(p,q)$ let $m=\gcd(k,p)$ . There exist $x_{j}^{\prime}\in\pi_{k}^{-1}(x)\subset S_{k}$ with $1\leq j\leq m$ and

$\pi_{k}^{-1}(o(x,g))=\sqcup_{j=1}^{m}o(x_{j}^{\prime},\tilde{g}_{k})$ (2.2)

the period of each $x_{j}^{\prime}$ under $\tilde{g}_{k}$ equal to $kq/m$ , and $T_{k}(x_{j}^{\prime})=x_{j+1}^{\prime}$ with indices $\mkern-8.0mu\mod k$ .

Proof.

To prove (a) we begin with two preliminary facts with similar proofs. First, we show that for any $z^{\prime}\in{\tilde{Z}}$ , $\pi_{k}({\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))})=Z$ . Let $z=\pi_{k}(z^{\prime})$ and pick $y\in Z$ . By minimality there exists $g^{n_{i}}(z)\rightarrow y$ . Lifting and using the compactness of $S_{k}$ there is a subsequence $n_{i^{\prime}}$ and a $y^{\prime}\in S_{k}$ with $\tilde{g}_{k}^{n_{i^{\prime}}}(z^{\prime})\rightarrow y^{\prime}$ . Thus $g_{k}^{n_{i^{\prime}}}(z)=\pi_{k}(\tilde{g}_{k}^{n_{i^{\prime}}}(z^{\prime}))\rightarrow\pi_{k}(y^{\prime})$ and so $y=\pi_{k}(y^{\prime})$ .

Second, we show that for any $z^{\prime},y^{\prime}\in{\tilde{Z}}$ , there exists an $p$ with $T^{p}(y^{\prime})\in{\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}$ . Let $z=\pi_{k}(z^{\prime})$ and $y=\pi_{k}(y^{\prime})$ . By minimality again, we have $g^{n_{i}}(z)\rightarrow y$ . Lifting and passing to a subsequence, there is a subsequence $n_{i^{\prime}}$ and a $y^{\prime\prime}\in S_{k}$ with $\tilde{g}_{k}^{n_{i^{\prime}}}(z^{\prime})\rightarrow y^{\prime\prime}$ . Thus $\pi_{k}(y^{\prime\prime})=y$ also, so there is a $p$ with $y^{\prime\prime}=T_{k}^{p}(y^{\prime})$ and so $T_{k}^{p}(y^{\prime})\in{\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}$ .

Now for the main proof, pick $z^{\prime}\in{\tilde{Z}}$ and let $Z_{1}^{\prime}={\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}$ , so by the first fact, $\pi_{k}(Z_{1}^{\prime})=Z$ . We now show $Z_{1}^{\prime}$ is minimal under $\tilde{g}_{k}$ . If not, there is a $y^{\prime}\in Z_{1}^{\prime}$ with ${\operatorname{Cl}(o(y^{\prime},\tilde{g}_{k}))}{\subsetneq}{\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}$ . By the second preliminary fact, there is some $p$ with

\operatorname{Cl}(o(T_{k}^{p}(z^{\prime}),\tilde{g}_{k}))\subset{\operatorname{Cl}(o(y^{\prime},\tilde{g}_{k}))}{\subsetneq}{\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}.

Acting by the homeomorphism $T_{k}^{p}$ and iterating the strict inclusions we have

{\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}=\operatorname{Cl}(o(T_{k}^{pk}(z^{\prime}),\tilde{g}_{k})){\subsetneq}\operatorname{Cl}(o(T_{k}^{p(k-1)}(z^{\prime}),\tilde{g}_{k})){\subsetneq}\dots\subsetneq\operatorname{Cl}(o(T_{k}^{p}(z^{\prime}),\tilde{g}_{k})){\subsetneq}{\operatorname{Cl}(o(z^{\prime},\tilde{g}_{k}))}

a contradiction, so $\tilde{g}_{k}$ acting on $Z_{1}^{\prime}$ is minimal. Thus since $\tilde{g}_{k}T_{k}=T_{k}\tilde{g}_{k}$ , $\tilde{g}_{k}$ acting on each $Z_{j}^{\prime}:=\tilde{g}_{k}^{j}(Z_{1}^{\prime})$ is minimal. Now minimal sets either coincide or are disjoint, so there is a least $m$ with $T_{k}^{m+1}Z_{1}^{\prime}=Z_{1}^{\prime}$ .

For the proof (b), assume $\pi_{k}(Z^{\prime})=\pi_{k}(Z^{\prime\prime})$ . Now $Z:=\pi_{k}(Z^{\prime})$ is minimal under $g$ and thus since $Z^{\prime}\subset\pi_{k}^{-1}(Z)$ and minimal sets are always disjoint or equal, using (2.1) we have that $Z^{\prime}=Z_{j}^{\prime}$ for some $j$ . Similarly, $Z^{\prime\prime}=Z_{j^{\prime}}^{\prime}$ for some $j^{\prime}$ , and thus $Z^{\prime}=T^{p}{Z_{1}^{\prime}}$ and $Z^{\prime\prime}=T^{p^{\prime}}{Z_{1}^{\prime}}$ and so $Z^{\prime\prime}=T^{p^{\prime}-p}(Z^{\prime})$ as required.

Now for (c), Since the deck group of $S_{k}$ is ${\mathbb{Z}}_{k}$ there is a natural identification of $\pi_{k}^{-1}(x)$ with ${\mathbb{Z}}_{k}$ with $x_{1}^{\prime}$ identified with zero. Since $\tilde{g}_{k}^{q}({\tilde{x}})=T^{p}{\tilde{x}}$ in ${\mathbb{R}}$ the induced action of $\tilde{g}_{k}^{q}$ on ${\mathbb{Z}}_{k}$ is $n\mapsto n+p\mod k$ . An easy elementary number theory argument yields that this action has exactly $\gcd(p,k)$ distinct orbits. Thus $\tilde{g}_{k}^{q}$ has exactly $\gcd(p,k)$ distinct orbits when acting on $\pi^{-1}_{k}(x)$ . But $x_{i}^{\prime},x_{j}^{\prime}\in\pi^{-1}_{k}(x)$ are on the same $\tilde{g}_{k}^{q}$ orbit if and only if they are on the same $\tilde{g}_{k}$ orbit and each orbit in $\pi^{-1}_{k}(o(x,g))$ contains at least one point from $\pi^{-1}_{k}(x)$ . Thus $\tilde{g}_{k}$ acting on $\pi^{-1}_{k}(o(x,g))$ has exactly $\gcd(p,k)$ orbits. The rest of the form of (2.2) follows from part (a). ∎

While it is not used here a similar result holds for ${\mathbb{Z}}$ covers and their cyclic quotients when a map in the base has a lift that commutes with the deck transformations.

Remark 2.2.

Some special cases of (c) are worth pointing out. If $\gcd(p,k)=1$ , then $x$ lifts to a single period $qk$ orbit in $S_{k}$ . If $p=k$ , then $x$ lifts to a $k$ different period- $q$ orbits in $S_{k}$ . When $k=2$ , there is a simple dichotomy. When $p$ is odd, $x$ lifts to one period $2q$ orbit and when $p$ is even, $x$ lifts to a pair of period $q$ orbits.

2.3. Rotation number and interval

For $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ a fixed lift of a degree one $g:S^{1}\rightarrow S^{1}$ define the rotation number of $x^{\prime}\in{\mathbb{R}}$ as

\rho(x^{\prime},\tilde{g})=\lim_{n\rightarrow\infty}\frac{\tilde{g}^{n}(x^{\prime})-x^{\prime}}{n}

(2.3)

when the limit exists. Note that this value depend in a simple way on the choice of lift $\tilde{g}$ of $g$ , namely, $\rho(x^{\prime},\tilde{g}+m)=\rho(x^{\prime},\tilde{g})+m$ . In most cases below there will be a preferred lift of a given $g$ that will be used in (2.3) and we define $\rho(x,g)=\rho(x^{\prime},\tilde{g})$ where $x^{\prime}$ is a lift of $x$ . When $g$ is understood we will just write $\rho(x)$ . If $x$ is a periodic point of rotation type $(p,q)$ then $\rho(x)=p/q$ .

If $Z$ is a $g$ -invariant set, let

\rho(Z)=\{\rho(x,g):x\in Z\}

and $\rho(g)=\rho(S^{1},g)$ . The latter set has been proved to be a closed interval [27, 38] and thus it is called the rotation interval of the map. We shall also have occasion to use $\rho(\tilde{g})$ with the obvious meaning.

There is a alternative way of computing the rotation interval using upper and lower maps that is now standard ([17, 36, 28, 13] and elsewhere). Given a lift of a degree-one circle map $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ let $\tilde{g}_{u}(x)=\sup\{\tilde{g}(y)\colon y\leq x\}$ and $\tilde{g}_{\ell}(x)=\inf\{\tilde{g}(y)\colon y\geq x\}$ . If $g_{u}$ and $g_{\ell}$ are their descents to $S^{1}$ they are both semi-monotone maps and so each of their rotation sets is a single point (see Lemma 3.1 below). The rotation interval of $g$ is

\rho(g)=[\rho(g_{\ell}),\rho(g_{u})].

(2.4)

To define the rotation number of a $g$ -invariant Borel probability measure $\mu$ start by letting $\Delta_{g}:S^{1}\rightarrow{\mathbb{R}}$ be $\Delta_{g}(x)=\tilde{g}(x^{\prime})-x^{\prime}$ which is independent of the choice of lift $x^{\prime}$ . Then

\rho(\mu)=\int\Delta_{g}\;d\mu

(2.5)

Note that when $\mu$ is ergodic by the Pointwise Ergodic Theorem for $\mu$ a.e. $x$ , $\rho(x,g)=\rho(\mu)$ .

For points, invariant sets and measures in the cyclic cover $S_{k}$ under the preferred lift $\tilde{g}_{k}$ , there are two ways to consider the rotation number. The most common will be to project to the base and define

\rho_{k}(x,g)=\rho(\pi_{k}(x),g)

(2.6)

For $\mu$ a $\tilde{g}_{k}$ invariant measure, let

\rho_{k}(\mu)=\rho((\pi_{k})_{*}\mu).

(2.7)

Remark 2.3.

The other way to work with rotation numbers in $S_{k}$ is to consider $\tilde{g}_{k}$ as a map of the circle itself. To work on the standard circle we first rescale $S_{k}$ via $D_{k}:S_{k}\rightarrow S^{1}$ via $D_{k}(\theta)=\theta/k$ . Note that $D_{k}$ is not a covering map but rather a coordinate rescaling homeomorphism. For $Z\subset S_{k}$ , then $\rho(D_{k}Z,D_{k}\circ\tilde{g}_{k}\circ D_{k}^{-1})$ is the desired rotation number. These two methods are related simply by $\rho_{k}(x,g)=k\rho(D_{k}Z,D_{k}\circ\tilde{g}_{k}\circ D_{k}^{-1})$

3. Semi-monotone degree-one maps

3.1. Definition and basic properties

In this section we give the basics of a small, but crucial expansion of the class of circle homeomorphisms, namely, continuous maps whose lifts are semi-monotone. They share many of the properties of circle homeomorphisms and are a standard and important tool in circle dynamics

Thus we consider continuous, degree one $h:S^{1}\rightarrow S^{1}$ whose lifts ${\tilde{h}}$ to ${\mathbb{R}}$ satisfy $x_{1}^{\prime}<x_{2}^{\prime}$ implies ${\tilde{h}}(x_{1}^{\prime})\leq{\tilde{h}}(x_{2}^{\prime})$ .¹¹1In topology a monotone map is one with connected point inverses. In this sense a semi-monotone map is monotone. On the other hand, considering the point of view of order relations, semi-monotone is contrast with monotone. We adapt the latter viewpoint. Note that this is independent of the choice of lift ${\tilde{h}}$ of $h$ . We shall also call such maps weakly order preserving. Let ${\mathcal{H}}$ be the collection of all such maps with the $C^{0}$ -topology, and $\tilde{{\mathcal{H}}}$ denotes all their lifts.

A flat spot for a $h\in{\mathcal{H}}$ is a nontrivial closed interval $J$ where $h(J)$ is a constant and for which there is no larger interval containing $J$ on which $h$ is constant. A given $h$ can have at most a countable number of flat spots $J_{i}$ and we define the “positive slope region” of $h$ as $P(h)=S^{1}\setminus(\cup\operatorname{Int}{J_{i}})$ . The proof of the next result is standard.

Lemma 3.1.

Assume $h\in{\mathcal{H}}$ with preferred lift ${\tilde{h}}$ .

(a)

The rotation number $\rho(x,h)$ exists and is the same for all $x\in S^{1}$ and so $\rho(h)$ is a single number.
(b)

The map $\rho:\tilde{{\mathcal{H}}}\rightarrow{\mathbb{R}}$ is continuous.
(c)

If ${\tilde{h}},{\tilde{h}}_{1}\in\tilde{{\mathcal{H}}}$ and ${\tilde{h}}_{1}\leq{\tilde{h}}$ then $\rho({\tilde{h}}_{1})\leq\rho({\tilde{h}})$ .
(d)

If $\rho(h)=p/q$ in lowest terms then all periodic orbits have rotation type $(p,q)$ and the recurrent set of $h$ consists of a union of such periodic orbits.
(e)

If $\rho(h)=\alpha\not\in{\mathbb{Q}}$ then $h$ has exactly one recurrent set which is a minimal set $Z$ and it is wholly contained in $P(h)$ . Further, $h$ is uniquely ergodic with the unique invariant measure supported on $Z$ .

Definition 3.2.

The minimal set in (e) above is called a semi-Denjoy minimal set. More generally, an abstract minimal set is called semi-Denjoy if it is topologically conjugate to the semi-Denjoy minimal set in a semi-monotone degree one circle map.

Remark 3.3.

A semi-Denjoy minimal set looks like a usual Denjoy minimal set with the added feature that endpoints of gaps can collapse to a point under forward iteration. It is clear that any $h\in{\mathcal{H}}$ is a near-homeomorphism (the uniform limit of homeomorphisms). Thus as a consequence of a theorem of Morton Brown ([14]), the inverse limit ${\varprojlim}(h,S^{1})$ is a circle and the natural extension is a circle homeomorphism. In particular, the inverse limit of a semi-Denjoy minimal set is a Denjoy minimal set. For example, in case of single flat spot, the two endpoints of the flat spot form a gap in the minimal set and they have same forward orbit. Taking the inverse limit splits open this orbit into a forward invariant gap.

3.2. Finitely many flat spots

We next introduce a subclass of ${\mathcal{H}}$ which includes the semi-monotone maps considered in this paper. Let ${\mathcal{H}}_{\ell}$ consist of those $h\in{\mathcal{H}}$ which have exactly $\ell$ flat spots and in $P(h)$ we require that $h$ is $C^{1}$ and $h^{\prime}>1$ where we have used a one-sided derivative at the end points of the flat spots.

Definition 3.4.

If $h\in{\mathcal{H}}_{\ell}$ and $\rho(h)\not\in{\mathbb{Q}}$ has semi-Denjoy minimal set $Z$ , since $Z\subset P(h)$ for any flat spot $J$ , $Z\cap\operatorname{Int}(J)=\emptyset$ . The flat spot $J$ is called tight for $h$ if $\operatorname{Fr}(J)\subset Z$ , and otherwise the flat spot is loose.

Lemma 3.5.

Assume $h\in{\mathcal{H}}_{\ell}$ .

(a)

If $\rho(h)=p/q$ in lowest terms then the number of $(p,q)$ -periodic orbits wholly contained in $P(h)$ is at least one and at most $\ell$ .
(b)

If $\rho(h)\not\in{\mathbb{Q}}$ , a flat spot $J_{i}$ is loose if and only if there is an $n>0$ and a $i^{\prime}$ with $h^{n}(J_{i})\in J_{i^{\prime}}$ . In particular, there is always at least one tight flat spot.
(c)

If $Z$ is the maximal recurrent set of $h$ in $P(h)$ , then

$Z=S^{1}\setminus\bigcup_{n=0}^{\infty}\bigcup_{i=1}^{\ell}h^{-n}(\operatorname{Int}(J_{i})),$ (3.1)

and so if $o(x,h)\subset P(h)$ then $h^{n}(x)\in Z$ for some $n\geq 0$ .

Proof.

For part (a), since $\rho(h)=p/q$ in lowest terms, every periodic point has period $q$ . By the conditions on the derivatives of $h\in{\mathcal{H}}_{\ell}$ , there are four classes of periodic points.

(1)

$x$ is unstable with $Dh^{q}(x)>1$ and $o(x,h)\subset\operatorname{Int}(P(h))$ .
(2)

$x$ is superstable with $Dh^{q}(x)=0$ and $o(x,h)\cap(\cup_{i=1}^{\ell}\operatorname{Int}(J_{i}))\not=\emptyset$ while $o(x,h)\cap(\cup_{i=1}^{\ell}\operatorname{Fr}(J_{i}))=\emptyset$ .
(3)

$x$ is superstable with $Dh^{q}(x)=0$ and $o(x,h)\cap(\cup_{i=1}^{\ell}\operatorname{Int}(J_{i}))\not=\emptyset$ while $o(x,h)\cap(\cup_{i=1}^{\ell}\operatorname{Fr}(J_{i}))=\emptyset$ and $o(x,h)$ contains both left and right endpoints of flat spots.
(4)

$x$ is semistable with $Dh^{q}(x)=0$ from one side and $Dh^{q}(x)>1$ from the other and $o(x,h)\cap(\cup_{i=1}^{\ell}\operatorname{Int}(J_{i}))=\emptyset$ while $o(x,h)\cap(\cup_{i=1}^{\ell}\operatorname{Fr}(J_{i}))\not=\emptyset$ and $o(x,h)$ contains only left or only right endpoints of flat spots.

This implies that all periodic points are isolated so there are finitely many of them.

Let $n_{i}$ be the number of periodic orbits of type (1). Using the fixed point index on $h^{q}$ , $n_{1}=n_{2}+n_{3}$ . Each orbit of type (3) hits two flat spots and each of type (2) and (4) at least one and a flat spot can’t contain multiple periodic orbits and so $n_{1}+2n_{3}+n_{4}\leq\ell$ . Thus the total number of periodic orbits wholly contained on $P(h)$ is $n_{1}+n_{3}+n_{4}=n_{2}+2n_{3}+n_{4}\leq\ell$

For part (b) assume first that $h^{n}(J_{i})\cap J_{i^{\prime}}=\emptyset$ for all $n>0$ and $i^{\prime}$ . If $J_{i}$ was loose, there would exist $z_{1},z_{2}\in Z$ with $z_{1}\leq J_{i}\leq z_{2}$ with at least one inequality strict and $Z\cap(z_{1},z_{2})=\emptyset$ . Thus $h((z_{1},z_{2}))$ is a nontrivial interval with $h^{n}((z_{1},z_{2}))\subset P(h)$ for all $n>0$ . This is impossible since $h$ is expanding in $P(h)$ and so $J_{i}$ must be loose.

For the converse, say $h^{n}(J_{i})\in J_{i^{\prime}}$ for some $n>0$ and first note $i=i^{\prime}$ is impossible since that would imply $h$ has a periodic point. Since $h^{n}(J_{i})$ is a point there exists a nontrivial interval $[z_{1},z_{2}]$ properly containing $J_{i}$ with $h^{n}([z_{1},z_{2}])=J_{i^{\prime}}$ and so $(z_{1},z_{2})\cap Z=\emptyset$ and so $J_{i}$ is a loose flat spot.

Finally, since $h^{n}(J_{i})\cap J_{i}=\emptyset$ and there are finitely many flat spots there is at least one $J_{i}$ with $h^{n}(J_{i})\cap J_{i^{\prime}}=\emptyset$ for all $n>0$ and $i^{\prime}$ .

For (c), assume $y$ is such that $o(y,h)\subset P(h)$ and $o(y,h)\cap Z=\emptyset$ . Let $x,x^{\prime}\in Z$ with $y\in(x,x^{\prime})$ and $(x,x^{\prime})\cap Z=\emptyset$ . Because of the uniform expansion in $P(h)$ there is a flat spot $J$ and an $n\geq 0$ so that $J\subset h^{n}([x,x^{\prime}])$ . If $\rho(h)\not\in{\mathbb{Q}}$ , then by (c) for some $n^{\prime}$ , $h^{n+n^{\prime}}([x,x^{\prime}])$ is a tight flat spot and so $h^{n+n^{\prime}+1}(y)\in Z$ .

Now assume $\rho(h)=p/q$ . In this case $x$ and $x^{\prime}$ are periodic orbits and so $J\subset h^{n+wq}([x,x^{\prime}])$ for all $w\geq 0$ and so $h^{n+wq}(y)\in h^{n+wq}([x,x^{\prime}])\setminus\operatorname{Int}(J)$ using the assumption that $o(y,h)\subset P(h)$ . But from (a), $h^{n+wq}(J)\subset J$ . Thus by monotonicity, $h^{n+wq}(y)$ is either always in the left component of $[x,x^{\prime}]\setminus\operatorname{Int}(J)$ or in the right component. This violates the expansion in $P(h)$ and so for some $j>0$ , $h^{j}(y)\in Z$ which yields (3.1). ∎

4. A class of bimodal circle maps and their positive slope orbits

4.1. The class ${\mathcal{G}}$

We introduce the class of bimodal, degree one maps of the circle that will be the focus here. The class is defined using properties of their lifts. We say that a lift $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is piecewise smooth if it is continuous and there are $0\leq x_{0}\leq\dots\leq x_{n}\leq 1$ so that $\tilde{g}$ is $C^{2}$ in each interval $(x_{i},x_{i+1})$ and the right and left hand derivatives exist at each $x_{i}$ .

Definition 4.1.

Let $\tilde{{\mathcal{G}}}$ be the class of all $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ with the following properties.

(a)

$\tilde{g}$ is piecewise smooth and $\tilde{g}(x^{\prime}+1)=\tilde{g}(x^{\prime})+1$ .
(b)

There are a pair of points $0={x_{min}}<{x_{max}}<1$ so that $g^{\prime}>1$ in $[{x_{min}},{x_{max}}]$ and $g$ is monotone decreasing in $[{x_{max}},{x_{min}}+1]$ .
(c)

${x_{min}}\leq\tilde{g}({x_{min}})<\tilde{g}({x_{max}})\leq{x_{max}}+1$

The class ${\mathcal{G}}$ consists of all $g:S^{1}\rightarrow S^{1}$ which have a lift in $\tilde{{\mathcal{G}}}$ .

Note that without loss of generality we have assumed that ${x_{min}}=0$ . Also by assumption, ${x_{min}}$ and ${x_{max}}$ are a nonsmooth local minimum and maximum respectively. It follows from (2.4) that $g\in{\mathcal{G}}$ implies $\rho(g)\subset[0,1]$ .

Standing assumption: From this point on $g$ denotes a given element of ${\mathcal{G}}$ and its preferred lift is the one with $\tilde{g}\in\tilde{{\mathcal{G}}}$ .

Remark 4.2.

Using the Parry-Milnor-Thurston Theorem for degree-one circle maps a general bimodal $h$ with $\rho(h)\subset(0,1)$ and not a point is semiconjugate to a PL map $g\in{\mathcal{G}}$ ²²2The result for circles doesn’t seem to be stated and proved anywhere in the literature, but as noted in [15] the proof in [1] works for the circle with minimal alteration. Point inverses of the semiconjugacy are either points or a closed interval. Thus using standard results from one-dimensional dynamics and various hypotheses most of the results of the paper can be transferred with appropriate alterations to a general bimodal map.

4.2. the model map

We will use a model map $f_{m}$ as a specific example throughout the paper. We shall see that, in a sense, it is the largest map in the class ${\mathcal{G}}$ and all other maps $g\in{\mathcal{G}}$ may be considered subsystems.

Define $\tilde{f}_{m}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ on $[0,1]$ as

\tilde{f}_{m}(x)=\begin{cases}3x&\text{for}\ 0\leq x\leq 1/2\\ -x+2&\text{for}\ 1/2\leq x\leq 1\end{cases}

and extend it to ${\mathbb{R}}$ to satisfy $\tilde{f}_{m}(x+1)=\tilde{f}_{m}(x)+1$ . Let $f_{m}$ be the projection of $\tilde{f}_{m}$ to $S^{1}$ . See Figure 2. Thus, ${x_{min}}=0$ , ${x_{max}}=1/2$ , and $\rho(f_{m})=[0,1]$ .

4.3. Positive slope orbits

Given $g\in{\mathcal{G}}$ with preferred lift $\tilde{g}$ let $\Lambda_{\infty}(g)$ be the points $x^{\prime}\in{\mathbb{R}}$ whose orbits under $\tilde{g}$ stay in the closed region where $\tilde{g}$ has positive slope, so

\Lambda_{\infty}(g)=\{x^{\prime}\in{\mathbb{R}}:o(x^{\prime},\tilde{g})\subset\bigcup_{j=-\infty}^{\infty}[j,j+{x_{max}}]\}.

We give $\Lambda_{\infty}(g)$ the total order coming from its embedding in ${\mathbb{R}}$ . Note that it is both $\tilde{g}$ and $T$ invariant.

Now we treat the $k$ -fold cover as $S_{k}=[0,k]/\mkern-8.0mu\sim$ and let $\Lambda_{k}(g)$ be the orbits that stay in the positive slope region of $\tilde{g}_{k}:S_{k}\rightarrow S_{k}$ , so

\Lambda_{k}(g)=\{x^{\prime}\in S_{k}:o(x^{\prime},\tilde{g}_{k})\subset\bigcup_{j=0}^{k-1}[j,j+{x_{max}}]\}.

Alternatively, $\Lambda_{k}(g)=p_{k}(\Lambda_{\infty}(g))$ or $\Lambda_{k}(g)=\pi_{k}^{-1}(\Lambda_{1}(g))$ .

We discuss the restriction to positive slope orbits in Section 13.2.

Standing assumption: Unless otherwise specified the terminology “physical kfsm set” or just “kfsm set” carries the additional restriction that it is contained in the positive slope region of some $g\in{\mathcal{G}}$ .

5. Symbolic description of positive slope orbits

For a map $g\in{\mathcal{G}}$ we develop in this section a symbolic coding for the orbits in $\Lambda_{k}$ for $k=1,\dots,\infty$ .

5.1. the itinerary maps

We work first in the universal cover or $k=\infty$ . Since $g\in{\mathcal{G}}$ , we may find points ${z_{max}}$ and ${z_{min}}$ with $0={x_{min}}<{z_{max}}<{z_{min}}<{x_{max}}$ and $\tilde{g}({z_{max}})={x_{max}}$ and $\tilde{g}({z_{min}})={x_{min}}+1$ . For $j\in{\mathbb{Z}}$ define a collection of intervals $\{I_{j}\}$ on ${\mathbb{R}}$ by

\begin{split}I_{2j}&=[j,{z_{max}}+j]\\ I_{2j+1}&=[{z_{min}}+j,{x_{max}}+j]\end{split}

(5.1)

See Figure 3. Note that since $\tilde{g}([{z_{max}},{z_{min}}])=[{x_{max}},{x_{max}}+1]$ we have that

\Lambda_{\infty}(g)=\{x^{\prime}\in{\mathbb{R}}:o(x^{\prime},\tilde{g})\subset\cup_{j=-\infty}^{\infty}I_{j}\}.

Using $\{I_{j}\}$ as an address system with the dynamics $\tilde{g}$ let the itinerary map be $\iota_{\infty}:\Lambda_{\infty}(g)\rightarrow\Sigma_{\mathbb{Z}}^{+}$ . Note that $\Lambda_{\infty}$ is the good set and using expansion and the disjointness of the address intervals, $\iota_{\infty}$ is a homemomorphism onto its image.

Now passing to the $k$ -fold cover, to code the positive slope orbits $\Lambda_{k}(g)$ , treat $S_{k}=[0,k]/\mkern-8.0mu\sim$ and use the dynamics $\tilde{g}_{k}$ with the address system $\{I_{0},I_{1},\dots,I_{2k-2},I_{2k-1}\}$ . This yields an itinerary map $\iota_{k}:\Lambda_{k}\rightarrow\Sigma_{2k}^{+}$ which is also a homeomorphism onto its image.

Example: The model map For the model map $f_{m}$ we have ${z_{max}}=1/6$ and ${z_{min}}=1/3$ and so $I_{2j}=[j,1/6+j]$ and $I_{2j+1}=[1/3+j,1/2+j]$

5.2. Symbolic analogs of covering spaces

This section develops the necessary machinery for the complete description of the image of the various itinerary maps. We will need the symbolic analogs of the covering spaces and maps described in Section 2.2.

Definition 5.1.

Define a subshift $\Omega_{\infty}\subset\Sigma_{\mathbb{Z}}^{+}$ by its allowable transitions

2j\rightarrow 2j,\ 2j\rightarrow 2j+1,\ 2j+1\rightarrow 2j+2,\ 2j+1\rightarrow 2j+3.

(5.2)

For $k<\infty$ let $\Omega_{k}$ be the subshift of $\Sigma_{2k}^{+}$ with allowable transitions as in (5.2) for $j=0,\dots,2k-1$ and indices reduced $\mkern-8.0mu\mod 2k$ .

Since for $g\in{\mathcal{G}}$ we have $\tilde{g}(I_{2j})\subset I_{2j}\cup I_{2j+1}$ and $\tilde{g}(I_{2j+1})\subset I_{2j+2}\cup I_{2j+3}$ we have:

Lemma 5.2.

For $g\in{\mathcal{G}}$ and $k=1,\dots,\infty$ , $\iota_{k}(\Lambda_{k}(g))\subset\Omega_{k}$ .

Under the itinerary maps the spaces ${\mathbb{R}}$ , $S_{k}$ and $S^{1}$ will correspond to the shift spaces $\Omega_{\infty}$ , $\Omega_{k}$ and $\Omega_{1}=\Sigma_{2}^{+}$ . The dynamics on the “physical spaces” induced by $g$ will correspond to left shifts on the symbol spaces. The shift spaces will also have the analogs of the covering projections and deck transformations. These maps will be indicated by a hat and defined using the action on individual symbols as follows.

The analogs of the covering translations are ${\hat{T}}_{\infty}:\Omega_{\infty}\rightarrow\Omega_{\infty}$ given by $s\mapsto s+2$ for all $s\in{\mathbb{Z}}$ and ${\hat{T}}_{k}:\Omega_{k}\rightarrow\Omega_{k}$ given by $s\mapsto s+2\mod 2k$ for all $s\in{\mathbb{Z}}$ , while the analogs of the covering maps are ${\hat{p}}_{k}:\Omega_{\infty}\rightarrow\Omega_{k}$ by $s\mapsto s\mod 2k$ and ${\hat{\pi}}_{k}:\Omega_{k}\rightarrow\Sigma_{2}^{+}$ by $s\mapsto s\mod 2$ . In the later we allow $k=\infty$ under the convention that $2\infty={\mathbb{Z}}$ , yielding ${\hat{\pi}}_{\infty}:\Omega_{\infty}\rightarrow\Sigma_{2}^{+}$ . Note then that ${\hat{\pi}}_{\infty}={\hat{p}}_{1}$ . A lift and the full lift are defined as usual with, for example, a lift of $Y\subset\Omega_{1}=\Sigma_{2}^{+}$ to $\Omega_{k}$ is a set $Y^{\prime}\subset\Omega_{k}$ with ${\hat{\pi}}_{k}(Y^{\prime})=Y$ . Note that ${\hat{T}}_{k},{\hat{\pi}}_{k}$ , and $p_{k}$ are all continuous.

The roles of the maps $g,\tilde{g}_{k}$ and $\tilde{g}$ in Section 2.2 are played by the various shift maps on the sequence spaces. For clarity we use a subscript to indicate which space the shift is acting on: $\sigma_{k}:\Omega_{k}\rightarrow\Omega_{k}$ . We again allow $k=\infty$ . All the various maps satisfy the same commutativity relations as their un-hatted analogs. So, for example, ${\hat{\pi}}_{k}{\hat{T}}_{k}=\pi_{k}$ , $\sigma_{k}{\hat{T}}_{k}={\hat{T}}_{k}\sigma_{k}$ , and ${\hat{\pi}}_{k}\sigma_{k}=\sigma_{1}{\hat{\pi}}_{k}$ . The itinerary maps $\iota_{k}:\Lambda_{k}(g)\rightarrow\Omega_{k}$ act naturally by transforming the spaces and maps of Section 2.2 to their symbolic analogs as in part (b) of the next Lemma.

Lemma 5.3.

For $k=1,\dots,\infty$ ,

(a)

$\Omega_{k}={\hat{p}}_{k}(\Omega_{\infty})$
(b)

${\hat{\pi}}_{k}\iota_{k}=\iota_{1}\pi_{k}$
(c)

If ${\underline{s}},{\underline{t}}\in\Omega_{k}$ and $\pi_{k}({\underline{s}})=\pi_{k}({\underline{t}})$ , then there exists a $n$ with ${\underline{s}}={\hat{T}}_{k}^{n}{\underline{t}}$ .

Proof.

Parts (a) and (b) are easy to verify. For (c) we prove the case $k=\infty$ which implies the $k<\infty$ cases. Assume ${\hat{\pi}}_{\infty}({\underline{s}})={\underline{w}}$ . The transitions in (5.2) coupled with the structure of ${\underline{w}}$ imply that once $s_{0}$ is set the parity structure of ${\underline{s}}$ determines all of ${\underline{w}}$ . Similarly, once $t_{0}$ is set all of ${\underline{t}}$ is determined. Once again (5.2) implies that if $s_{0}-t_{0}=2n$ then for all $i$ , $s_{i}=t_{i}+2n$ . ∎

Remark 5.4.

It would perhaps seem more natural that $\Sigma^{+}_{\mathbb{Z}}$ should act as the symbolic universal cover of $\Sigma_{2}^{+}$ , but the crucial covering space property expressed by (c) wouldn’t hold in this case. For example, if ${\underline{s}}=.131^{\infty}$ and ${\underline{t}}=.151^{\infty}$ then ${\hat{\pi}}_{\infty}({\underline{s}})={\hat{\pi}}_{\infty}({\underline{t}})$ but ${\hat{T}}^{n}({\underline{s}})\not={\underline{t}}$ for all $n$ .

5.3. Rotation numbers and sets

We give the analogs of the definitions in Section 2.3 for the symbolic case. For ${\underline{s}}\in\Sigma_{2}^{+}$ let

{\hat{\rho}}({\underline{s}})=\lim_{n\rightarrow\infty}\frac{1}{n+1}\sum_{i=0}^{n}s_{i}.

(5.3)

when the limit exists. For ${\hat{\mu}}$ a shift invariant measure on $\Sigma_{2}^{+}$ , let $\rho({\hat{\mu}})={\hat{\mu}}([1])$ . When ${\hat{\mu}}$ is ergodic, by the Pointwise Ergodic Theorem, for $\mu$ a.e. ${\underline{s}}$ , ${\hat{\rho}}({\underline{s}})={\hat{\rho}}(\mu)$

For ${\hat{Z}}\in\Omega_{k}$ let ${\hat{\rho}}_{k}({\hat{Z}})={\hat{\rho}}({\hat{\pi}}_{k}({\hat{Z}}))$ and for ${\hat{\mu}}$ a $\sigma_{k}$ invariant measure on $\Omega_{k}$ , let ${\hat{\rho}}_{k}({\hat{\mu}})={\hat{\rho}}(({\hat{\pi}}_{k})_{*}({\hat{\mu}}))$ .

6. Topological conjugacies and the image of the itinerary maps

In this section we develop the analog of kneading invariants for the symbolic coding of the positive slope orbits for $g\in{\mathcal{G}}$ .

Recall that $\Sigma_{2}^{+}$ is given the lexicographic order. Assume ${\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\in\Sigma_{2}^{+}$ satisfy

{\underline{\kappa}}_{0}\leq o({\underline{\kappa}}_{i},\sigma)\leq{\underline{\kappa}}_{1}

(6.1)

for $i=0,1$ . The corresponding dynamical order interval is

\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle=\{{\underline{s}}\colon{\underline{\kappa}}_{0}\leq o({\underline{s}},\sigma)\leq{\underline{\kappa}}_{1}\}

Returning to $g\in{\mathcal{G}}$ , note that $\tilde{g}(I_{0})\subset[{x_{min}},{x_{max}}]$ and $\tilde{g}(I_{1})\subset[{x_{min}},{x_{max}}]$ while $\tilde{g}([{z_{max}},{z_{min}}])=[{x_{max}},{x_{min}}+1]$ . This implies that $\Lambda_{1}(g)\subset[{x_{min}},{x_{max}}]$ . Since $\Lambda_{1}(g)$ is compact we may define ${\underline{\kappa}}_{0}={\underline{\kappa}}_{0}(g)=\iota_{1}(\min(\Lambda_{1}))$ and ${\underline{\kappa}}_{1}={\underline{\kappa}}_{1}(g)=\iota_{1}(\max(\Lambda_{1}))$ . By construction these ${\underline{\kappa}}$ ’s satisfy (6.1).

We showed above that $\iota_{k}(\Lambda(g))\subset\Omega_{k}$ . The next theorem says that the image is as constrained by the dynamical order interval $\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ . Accordingly for $k=1,\dots,\infty$ we define ${\hat{\Lambda}}_{k}(g)=\Omega_{k}\cap{\hat{\pi}}_{k}^{-1}(\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle)$ and note that this is a $\sigma_{k}$ invariant set.

Theorem 6.1.

Assume $g\in{\mathcal{G}}$ and construct $\kappa_{0}$ and $\kappa_{1}$ from $g$ as above. Then for $k=1,\dots,\infty$ the itinerary map $\iota_{k}$ is a topological conjugacy from $(\Lambda_{k}(g),(\tilde{g}_{k})_{|\Lambda_{k}(g)})$ to $({\hat{\Lambda}}_{k}(g),\sigma_{k})$ . Further, $\iota_{\infty}$ is order preserving.

Proof.

We first prove the first assertion for $k=1$ or that $\iota_{1}(\Lambda_{1}(g))=\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ . Let $\ast$ be an arbitrary symbol and define a map $\chi:[0,{x_{max}}]\sqcup\{\ast\}\rightarrow[0,{x_{max}}]\sqcup\{\ast\}$ by

\chi(x)=\begin{cases}\tilde{g}(x)\ \text{for}\ x\in I_{0}\\ \ast\ \text{for}\ x\in({z_{min}},{z_{max}})\sqcup\{\ast\}\\ \tilde{g}(x)-1\ \text{for}\ x\in I_{1}\end{cases}

It easily follow that

\Lambda_{1}(g)=\{x\in[0,{x_{max}}]\colon\chi^{n}(x)\not=\ast\ \text{for all}\ n>0\}.

and if use the dynamics $\chi$ with the address system $I_{0},I_{1}$ the resulting itinerary map $\Lambda_{1}\rightarrow\Sigma_{2}^{+}$ is exactly $\iota_{1}$ . Now since $g$ is expanding on $I_{0}\cup I_{1}$ and $I_{0}\cap I_{1}=\emptyset$ , $\iota_{1}$ is an order preserving conjugacy from $(\Lambda_{1},g)$ to $(\iota_{1}(\Lambda_{1}),\sigma_{1})$ . Finally, since $\min\Lambda_{1}(g)\leq o(x,g)\leq\max\Lambda_{1}(g)$ for all $x\in\Lambda_{1}(g)$ we have that $\kappa_{0}\leq o({\underline{s}},\sigma)\leq\kappa_{1}$ for all ${\underline{s}}\in\iota_{1}(\Lambda_{1})$ and further that for any such ${\underline{s}}$ there is an $x\in\Lambda_{1}(g)$ with $\iota_{1}(x)={\underline{s}}$ . Thus $\iota_{1}(\Lambda_{1}(g))=\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ .

We now show that

\iota_{k}(\Lambda_{k}(g))=\Omega_{k}\cap{\hat{\pi}}_{k}^{-1}(\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle)

(6.2)

We already know from Lemma 5.2 that the left hand side is in $\Omega_{k}$ . Next, since $\pi_{k}(\Lambda_{k}(g))=\Lambda_{1}(g)$ using Fact 5.3(b) and the first paragraph of the proof we have

\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle=\iota_{1}(\Lambda_{1}(g))=\iota_{1}(\pi_{k}(\Lambda_{k}(g)))={\hat{\pi}}_{k}\iota_{k}(\Lambda_{k}(g))

(6.3)

so the left hand side of (6.2) is also in ${\hat{\pi}}_{k}^{-1}(\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle)$ .

Now assume that ${\underline{s}}$ is in the right hand side of (6.2). Certainly then ${\hat{\pi}}_{k}({\underline{s}})\in\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ and so there is an $x\in\Lambda_{1}(g)$ with $\iota_{1}(x)={\hat{\pi}}_{k}({\underline{s}})$ . Pick a lift $x^{\prime}\in\Lambda_{k}(g)$ with $\pi_{k}(x^{\prime})=x$ . Again using Lemma 5.3(b)

{\hat{\pi}}_{k}({\underline{s}})=\iota_{1}(x)=\iota_{1}\pi_{k}(x^{\prime})={\hat{\pi}}_{k}\iota_{k}(x^{\prime}).

(6.4)

Thus using Lemma 5.3(c), there is an $n$ with $\iota_{k}(x^{\prime})={\hat{T}}_{k}^{n}({\underline{s}})$ and so

\iota_{k}{\hat{T}}_{k}^{-n}x^{\prime}={\hat{T}}_{k}^{-n}\iota_{k}x^{\prime}={\underline{s}}

and ${\hat{T}}_{k}^{-n}x^{\prime}\in\Lambda_{k}$ . Thus ${\underline{s}}\in\iota_{k}(\Lambda_{k})$ as required.

For $\iota_{\infty}$ as with $\iota_{1}$ , since the $I_{j}$ are disjoint and the $\tilde{g}_{|}I_{j}$ are expanding, we have that $\iota_{\infty}$ is an order preserving homeomorphisms onto its image. The fact that it is a semiconjugacy follows because it is an itinerary map. ∎

Example: The model map For the model map $f_{m}$ we have $\kappa_{0}=.0^{\infty}$ and $\kappa_{1}=.1^{\infty}$ and so in this case ${\hat{\Lambda}}_{k}(\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle)$ is the entire subshift $\Omega_{k}$ .

Remark 6.2.

(a)

${\hat{\rho}}\circ\iota_{k}=\rho$ (when defined) and ${\hat{\rho}}_{k}\circ\iota_{k}=\rho_{k}$
(b)

For $\mu$ a $g$ invariant measure supported in $\Lambda_{1}(g)$ , $\rho(\mu)=\mu(I_{1})$

7. $k$ -fold semi-monotone sets

While our eventual interest is in invariant sets in the circle, it is convenient to first give definitions in the universal cover ${\mathbb{R}}$ and the cyclic covers $S_{k}$ .

7.1. Definitions

The next definition makes sense for any degree one map but for concreteness we restrict to $g\in{\mathcal{G}}$ .

Definition 7.1.

Let $g\in{\mathcal{G}}$ have preferred lift $\tilde{g}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ .

(a)

A $\tilde{g}$ -invariant set $Z^{\prime}\subset{\mathbb{R}}$ is k-fold semi-monotone (kfsm) if $T^{k}(Z^{\prime})=Z^{\prime}$ and $\tilde{g}$ restricted to $Z^{\prime}$ is weakly order preserving, or for $z_{1}^{\prime},z_{2}^{\prime}\in Z^{\prime}$

$z_{1}^{\prime}<z_{2}^{\prime}\ \ \text{implies}\ \ \tilde{g}(z_{1}^{\prime})\leq\tilde{g}(z_{2}^{\prime})$
(b)

A $\tilde{g}_{k}$ -invariant set $Z\subset S_{k}$ is k-fold semi-monotone (kfsm) if it has a $\tilde{g}$ -invariant lift $Z^{\prime}\subset{\mathbb{R}}$ which is.

These definitions are independent of the choice of lift $\tilde{g}$ . Note that the same terminology is used for sets in the universal and cyclic covers and that implicit in being a kfsm set is the fact that the set is invariant.

When $k=1$ the lift $Z^{\prime}$ in the definition must satisfy $T(Z^{\prime})=Z^{\prime}$ and $\pi(Z^{\prime})=Z$ and so $Z^{\prime}=\pi^{-1}(Z)$ , the full lift to ${\mathbb{R}}$ .

7.2. interpolation

To say that $Z\subset S_{k}$ is $k$ -fold semi-monotone means roughly that it is semi-monotone treating $S_{k}$ as the usual circle. To formalize this as in Remark 2.3 it will be useful to rescale $S_{k}$ to $S^{1}$ using $D_{k}:S_{k}\rightarrow S^{1}$ and consider the map $D_{k}\circ\tilde{g}_{k}\circ D_{k}^{-1}$ .

Lemma 7.2.

The following are equivalent

(a)

The $\tilde{g}_{k}$ -invariant set $Z\subset S_{k}$ is kfsm
(b)

$DZ$ is $1$ -fold semi-monotone under $D_{k}\circ\tilde{g}_{k}\circ D_{k}^{-1}$ and there exists a semi-monotone circle map $h$ defined on $S_{k}$ which interpolates $\tilde{g}_{k}$ acting on $Z$ .
(c)

The lift $Z^{\prime}\subset{\mathbb{R}}$ of $Z$ in Definition 7.1(b) has the property that there is a continuous $H:{\mathbb{R}}\rightarrow{\mathbb{R}}$ that interpolates $\tilde{g}$ acting on ${\tilde{Z}}^{*}$ , is weakly order preserving, and satisfies $H(x+k)=H(x)+k$ .

We now restrict to positive slope orbits as in Section 4.3 and collect together kfsm invariant sets in $S_{k}$ and their invariant measures. We will comment on kfsm sets which intersect the negative slope region in Section 13.2. We also restrict attention to invariant sets that are recurrent.

Definition 7.3.

Given $g\in{\mathcal{G}}$ let ${\mathcal{B}}_{k}(g)$ be all compact, recurrent kfsm sets in $\Lambda_{k}(g)\subset S_{k}$ with the Hausdorff topology and ${\mathcal{N}}_{k}(g)$ be all $\tilde{g}_{k}$ -invariant, Borel probability measures with the weak topology whose support is a $Z\in{\mathcal{B}}_{k}(g)$ .

Remark 7.4.

A standard argument from Aubry-Mather theory yields that the collection of all kfsm sets is compact in the Hausdorff topology. Since $\Lambda_{k}(g)$ is compact, the collection of positive slope kfsm sets is also compact. However, since ${\mathcal{B}}_{k}(g)$ contains just the recurrent kfsm sets, it is not compact (see Section 9.1 and 13.2. We show shortly that ${\mathcal{N}}_{k}(g)$ is compact.

7.3. symbolic k-fold semi-monotone sets and the map $g$

As with kfsm sets in the “physical” spaces $S_{k}$ and ${\mathbb{R}}$ we define their symbolic analogs in the symbol spaces $\Omega_{k}$ and $\Omega_{\infty}$ where we give the symbol spaces the lexicographic order.

Definition 7.5.

(1)

A $\sigma_{\infty}$ -invariant set ${\hat{Z}}^{\prime}\subset\Omega_{\infty}$ is symbolic k-fold semi-monotone (kfsm) if ${\hat{T}}_{\infty}^{k}({\hat{Z}}^{\prime})={\hat{Z}}^{\prime}$ and $\sigma_{\infty}$ restricted to ${\hat{Z}}^{\prime}$ is weakly order preserving, or for ${\underline{s}},{\underline{t}}\in{\hat{Z}}^{\prime}$

${\underline{s}}<{\underline{t}}\ \ \text{implies}\ \ \sigma_{\infty}({\underline{s}})\leq\sigma_{\infty}({\underline{t}}).$
(2)

A $\sigma_{k}$ -invariant set ${\hat{Z}}\subset\Omega_{k}$ is symbolic k-fold semi-monotone (kfsm) if there is a $\sigma_{\infty}$ -invariant lift ${\hat{Z}}^{\prime}$ to $\Omega_{\infty}$ . (i.e., ${\hat{p}}_{k}({\hat{Z}}^{\prime})={\hat{Z}}$ ) which is kfsm.

Everything has been organized thus far to ensure that k-fold semi-monotone sets are preserved under the itinerary maps.

Theorem 7.6.

Given $g\in{\mathcal{G}}$ for $k=1,2,\dots,\infty$ , a $\tilde{g}_{k}$ invariant set $Z\subset\Lambda_{k}(g)$ is kfsm if an only if $\iota_{k}(Z)\subset{\hat{\Lambda}}_{k}(g)$ is.

Proof.

We prove the $k=\infty$ case; the $k<\infty$ case follows. Theorem 6.1 shows that $\iota_{\infty}$ is an order preserving bijection. Since $\iota_{\infty}T_{\infty}^{k}={\hat{T}}_{\infty}^{k}\iota_{\infty}$ we have that $T_{\infty}^{k}(Z)=Z$ if and only if ${\hat{T}}^{k}_{\infty}\iota_{\infty}(Z)=\iota_{\infty}(Z)$ . Using the additional fact that $\iota_{\infty}\tilde{g}=\sigma_{\infty}\iota_{\infty}$ we have that $\tilde{g}$ is weakly order preserving on $Z$ if and only if $\sigma_{\infty}$ is weakly order preserving on $\iota_{\infty}(Z)$ ∎

In analogy with Definition 7.3 we collect together the various symbolic kfsm sets and their invariant measures.

Definition 7.7.

For $k<\infty$ given $g\in{\mathcal{G}}$ , let ${\hat{{\mathcal{B}}}}_{k}(g)$ be all compact, invariant, recurrent symbolic kfsm sets in ${\hat{\Lambda}}_{k}(g)$ with the Hausdorff topology and ${\hat{{\mathcal{N}}}}_{k}(g)$ be all $g$ -invariant, Borel probability measures with the weak topology whose support is a ${\hat{Z}}\in{\hat{{\mathcal{B}}}}_{k}(g)$ .

Lemma 7.8.

For $k<\infty$
(a)

The map $\iota_{k}:\Lambda_{k}(g)\rightarrow{\hat{\Lambda}}_{k}(g)$ induces homeomorphisms ${\mathcal{B}}_{k}(g)\rightarrow{\hat{{\mathcal{B}}}}_{k}(g)$ and ${\mathcal{N}}_{k}(g)\rightarrow{\hat{{\mathcal{N}}}}_{k}(g)$ .
(b)

The spaces ${\mathcal{N}}_{k}(g)$ and ${\hat{{\mathcal{N}}}}_{k}(g)$ are compact.

Proof.

For part (a) we know that $\iota_{k}$ is a conjugacy that that takes kfsm sets to kfsm sets which yields that ${\mathcal{N}}_{k}(g)\rightarrow{\hat{{\mathcal{N}}}}_{k}(g)$ is a homeomorphism. By hypothesis any $g\in{\mathcal{G}}$ is $C^{2}$ in $P(g)$ and so there is some $M>1$ with $g^{\prime}<M$ on $P(g)$ and thus on all address intervals $I_{j}$ . It is standard that this implies that $\iota_{k}$ is Hölder with exponent $\nu=\log 2k/\log M$ . This then implies that $\iota_{k}$ preserves Hausdorff convergence and so ${\mathcal{B}}_{k}(g)\rightarrow{\hat{{\mathcal{B}}}}_{k}(g)$ is a homeomorphism.

For part (b), since the space of all $\tilde{g}_{k}$ invariant Borel probability measures is compact metric, it suffices to show that ${\mathcal{N}}_{k}(g)$ is closed, and so assume $\mu_{n}\in{\mathcal{N}}_{k}(g)$ and $\mu_{n}\rightarrow\mu$ weakly with $X_{n}:=\operatorname{spt}(\mu_{n})$ a recurrent kfsm set.

A noted in Remark 7.4 the collection of all kfsm sets in $\Lambda_{k}$ is compact in the Hausdorff topology and so there exists a kfsm set $X$ and $n_{i}\rightarrow\infty$ with $X_{n_{i}}\rightarrow X$ . A standard argument which we give here shows that $\operatorname{spt}(\mu)\subset X$ . If this inclusion does not hold, there exists an $x\in\operatorname{spt}(\mu)\cap X^{c}$ , then let $\epsilon=d(x,X)$ . Since the atoms of $\mu$ are countable, we may find an $\epsilon_{1}<\epsilon/4$ so that letting $U=N_{\epsilon_{1}}(x)$ we have so that $\mu(\operatorname{Fr}(U))=0$ and so $U$ is a continuity set for $\mu$ thus via a standard result (page 16-17 of [6]) $\mu_{n_{i}}(U)\rightarrow\mu(U)>0$ using the fact that $x\in\operatorname{spt}(\mu)$ . Thus for large enough $i$ , with $m=n_{i}$ we have $X_{m}\subset N_{\epsilon/4}(X)$ and so $\emptyset=U\cap X_{m}=U\cap\operatorname{spt}(\mu_{m})$ with $\mu_{m}(U)>0$ a contradiction. Thus $\operatorname{spt}(\mu)\subset X$ . Now any invariant measure supported on $X$ must be supported on its recurrent set and so $\mu\in{\mathcal{N}}_{k}(g)$ , as required. The compactness of ${\hat{{\mathcal{N}}}}_{k}(g)$ follows from part(a). ∎

Example: The model map For the model map $f_{m}$ , ${\hat{\Lambda}}_{k}(f_{m})=\Omega_{k}$ , and so the set ${\hat{{\mathcal{B}}}}_{k}(f_{m})$ is the collection of all symbolic recurrent kfsm sets in $\Omega_{k}$ . Thus while the definition of symbolic kfsm set is abstract and general by Theorem 7.6 and Theorems 6.1 they will share all the properties of “physical” kfsm sets.

7.4. Rotation numbers and sets

For $Z\in{\mathcal{B}}_{k}(g)$ recall from section 2.3 that $\rho_{k}(Z)=\rho(\pi_{k}(Z),g)$ .

Lemma 7.9.

Assume $Z\in{\mathcal{B}}_{k}(g)$ ,

(a)

$\rho_{k}(Z)$ exists and is a single number.
(b)

If $\rho_{k}(Z)=\omega\not\in{\mathbb{Q}}$ then $Z$ is a semi-Denjoy minimal set.
(c)

If $\rho_{k}(Z)=p/q$ with $\gcd(p,q)=1$ , then $Z$ consists of at least one and at most $k$ periodic orbits all with the same rotation number and period equal to $qk/\gcd(p,k)$ .
(d)

$\rho_{k}:{\mathcal{B}}_{k}(g)\rightarrow{\mathbb{R}}$ and ${\hat{\rho}}_{k}:{\hat{{\mathcal{B}}}}_{k}(g)\rightarrow{\mathbb{R}}$ are continuous

Proof.

By Theorem 7.2 there exists a continuous, semi-monotone $H:S_{k}\rightarrow S_{k}$ which interpolates the action of $\tilde{g}_{k}$ on $Z$ . Rescaling to the standard circle let $H_{k}:S^{1}\rightarrow S^{1}$ be defined as $H_{k}:=D_{k}\circ H\circ D_{k}^{-1}$ . By Lemma 3.1(a), $\rho(H_{k})=\omega$ is a single number and since $\rho_{k}(Z)=k\rho(DZ,H_{k})$ , (a) follows. If $\rho_{k}(Z)\not\in{\mathbb{Q}}$ then $\rho(DZ,H_{k})\not\in{\mathbb{Q}}$ and so by Lemma 3.1(e) $DZ$ and thus $Z$ is a semi-Denjoy minimal set yielding (b).

Now assume $\rho_{k}(Z)=p/q$ in lowest terms and so $\rho(DZ,H_{k})=p/(qk)$ . Written in lowest terms

\frac{p}{qk}=\frac{p/\gcd(p,k)}{kq/\gcd(p,k)}.

But since $H_{k}$ is semi-monotone, its recurrent set is a collection of periodic orbits and its rotation number in lowest terms has their period as its denominator which is thus $qk/\gcd(p,k)$ . Since by assumption, $Z\subset\Lambda_{k}(g)$ we may choose $H$ to have $k$ flat spots then using Lemma 3.5, $Z$ consists of at least one and at most $k$ periodic orbits, finishing (c).

It is standard from Aubry-Mather theory that $\rho$ is continuous on the collection of all kfsm sets and thus it is continuous restricted to the recurrent kfsm sets. As for measures, since $\rho(\mu)=\int\Delta_{g}\;d\mu$ (2.5) with $\Delta_{g}$ continuous, continuity follows from the definition of weak convergence. ∎

Definition 7.10.

If $Z\subset{\mathcal{B}}_{k}$ and consists of a finite collection of periodic orbits it is called a cluster.

Remark 7.11.

(a)

For the case of general recurrent symbolic kfsm ${\hat{Z}}$ as we commented at the end of the last subsection we may consider ${\hat{Z}}\in{\hat{\Lambda}}_{k}(f_{m})=\Omega_{k}$ with $f_{m}$ the model map. Using the itinerary map $\iota_{k}:\Lambda_{k}(f_{m})\rightarrow{\hat{\Lambda}}_{k}(f_{m})$ we have from Theorem 7.6 that $(\iota_{k})^{-1}({\hat{Z}})$ is kfsm for $f_{m}$ and then all the conclusions of the previous theorem hold for it. Then using Theorem 6.1, the conclusions of the previous theorem hold with the obvious addition of hats in the appropriate places.
(b)

We shall need this implication of the symbolic case below. If ${\hat{Z}}\subset\Omega_{k}$ with $\rho_{k}({\hat{Z}})=\alpha\not\in{\mathbb{Q}}$ , tnen there exists a continuous, onto $\phi:{\hat{Z}}\rightarrow S_{k}$ which is weakly order preserving, $\phi\sigma_{k}=R_{\alpha}\phi$ , and $\#\phi^{-1}(x)=1$ for all but a countable number of $R_{\alpha}$ -orbits on which $\#\phi^{-1}(x)=2$ .
(c)

Using Lemma 3.1 a measure in ${\mathcal{N}}_{k}(g)$ is either the unique measure on a semi-Denjoy minimal set or a convex combination or measures supported on the periodic orbits in a cluster.
(d)

A $Z\in{\mathcal{B}}_{k}(g)$ is minimal if and only if it is uniquely ergodic and similarly for $Z\in{\hat{{\mathcal{B}}}}_{k}(g)$

8. The HM construction

For a given $g\in{\mathcal{G}}$ at this point we have reduced the identification of its positive slope kfsm sets to a question in symbolic dynamics. In this section we answer this symbolic question via a generalization of the procedure of Hedlund and Morse. The generalization constructs all symbolic kfsm recurrent sets for each $k$ .

Since a linear order is essential to the notion of semi-monotone we will again begin working on the line and then project to cyclic covers.

8.1. definition and basic properties

Fix an integer $k>0$ , a real number $\omega\in(0,1)$ , and a vector $\vec{\nu}=(\nu_{1},\dots,\nu_{k})$ with $\nu_{i}\geq 0$ and $\sum\nu_{i}=k-k\omega$ . Such a pair $(\omega,\vec{\nu})$ is called allowable. Start with the intervals defined for $0\leq j\leq k-1$ by

\begin{split}X_{2j}&=\left(\sum_{i=1}^{j}\nu_{i}+j\omega,\sum_{i=1}^{j+1}\nu_{i}+j\omega\right)\\ X_{2j+1}&=\left(\sum_{i=1}^{j+1}\nu_{i}+j\omega,\sum_{i=1}^{j+1}\nu_{i}+(j+1)\omega\right)\end{split}

(8.1)

and then extend for $\ell\in{\mathbb{Z}}$ and $0\leq m\leq 2k-1$ as $X_{\ell k+m}=X_{m}+\ell k$ . Thus each $X_{2j}$ has width $\nu_{j+1}$ and each $X_{2j+1}$ has width $\omega$ and the entire structure yields a $T^{k}$ invariant address system under the dynamics $R_{\omega}(x)=x+\omega$ on ${\mathbb{R}}$

The good set is $G_{\infty}$ depends on $k,\omega$ and $\nu$ and is given by

G=\{x^{\prime}\in{\mathbb{R}}:o(x^{\prime},R_{\omega})\cap\partial X_{i}=\emptyset\ \text{for all}\ i\}

Note that $G$ is dense, $G_{\delta}$ and has full Lebesgue measure. The itinerary map with respect to the given address system is denoted $\zeta_{\infty}:G\rightarrow\Sigma^{+}_{{\mathbb{Z}}}$ .

Definition 8.1.

Let $A_{k}(\omega,\vec{\nu})=\operatorname{Cl}(\zeta_{\infty}(G))$ .

Remark 8.2.

By construction, $A_{k}(\omega,\vec{\nu})$ is $\sigma_{\infty}$ and ${\hat{T}}_{\infty}^{k}$ invariant. In addition, since for all $j$ , $R_{\omega}(X_{2j})\subset X_{2j}\cup X_{2j+1}$ and $R_{\omega}(X_{2j+1})\subset X_{2j+2}\cup X_{2j+3}$ we have $\zeta_{\infty}(G_{\infty})\subset\Omega_{\infty}$ . Also, since $\Omega_{\infty}$ is compact, $A_{k}(\omega,\vec{\nu})\subset\Omega_{\infty}$ is also.

8.2. Cyclic covers

We now return to the compact quotients where the recurrent dynamics takes place and introduce measures into the HM-construction.

For fixed $k>0$ and allowable $(\omega,\vec{\nu})$ treat $\{X_{0},\dots,X_{2k-1}\}\subset S_{k}=[0,k]/\mkern-8.0mu\sim$ as an address system under the dynamics given by $R_{\omega}(x)=x+\omega\mod k$ . Define the good set $G_{k\omega\vec{\nu}}$ and on it define the itinerary map $\zeta_{k\omega\vec{\nu}}$ . We will often suppress the dependence of these quantities on various of the subscripted variables when they are clear from the context.

Definition 8.3.

Given $k$ and an allowable $(\omega,\vec{\nu})$ define the itinerary map $\zeta_{k}:G_{k}\rightarrow\Sigma^{+}_{2k}$ as above. Let $B_{k}(\omega,\vec{\nu})=\operatorname{Cl}(\zeta_{k}(G_{k}))\subset\Sigma_{2k}^{+}$ and $\lambda_{k}(\omega,\vec{\nu})=(\zeta_{k})_{*}(m)/k$ where $m$ is the measure on $S_{k}$ induced by Lebesgue measure on ${\mathbb{R}}$ .

Remark 8.4.

(1)

By construction ${\hat{p}}_{k}(A_{k}(\omega,\vec{\nu}))=B_{k}(\omega,\vec{\nu})\subset\Omega_{k}$ . and so $\rho_{k}(B_{k}(\omega,\vec{\nu}))=\omega$ .
(2)

Let $W_{k}=\{(x,\omega,\vec{\nu})\colon x\in G_{k\omega\vec{\nu}}\}.$ It is easy to check that the map $(x,\omega,\vec{\nu})\mapsto\zeta_{k\omega\vec{\nu}}(x)$ is continuous on $W_{k}$ .

The next theorem describes the structure of the $B_{k}(\omega,\vec{\nu})$ and shows that all symbolic kfsm sets are constructed by the HM procedure with $\omega$ equal to their rotation number.

Theorem 8.5.

(a)

For $\alpha\not\in{\mathbb{Q}}$ , $B_{k}(\alpha,\nu)$ is a semi-Denjoy minimal set with unique invariant probability measure $\lambda_{k}(\omega,\vec{\nu})$ .
(b)

For $p/q\in{\mathbb{Q}}$ , $B_{k}(p/q,\nu)$ is a finite collection of periodic orbits each with rotation number $p/q$ and period $qk/\gcd(p,k)$ , and $\lambda_{k}(p/q,\nu)$ is a convex combination of the measures supported on the periodic orbits.
(c)

A ${\hat{Z}}\subset\Omega_{k}$ is a recurrent symbolic kfsm set with $\rho_{k}(Z)=\omega$ if and only if ${\hat{Z}}=B_{k}(\omega,\vec{\nu})$ for some allowable $\vec{\nu}$ . Thus the collection of invariant probability measures supported on symbolic recurrent kfsm set is exactly the collection of $\lambda_{k}(\omega,\vec{\nu})$ for all allowable $(\omega,\vec{\nu})$ .

Proof.

We begin by proving portions of (a) and (b). For part (a) we first show that $B_{k}(\alpha,\nu)$ is minimal using a characterization usually attributed to Birkhoff: If $f:X\rightarrow X$ is a continuous function of a compact metric space and $x\in X$ , then $\operatorname{Cl}(o(x,f))$ is a minimal set if and only if for all $\epsilon>0$ there exists an $N$ so that for all $n\in{\mathbb{N}}$ there is a $0<i\leq N$ with $d(f^{n+i},x)<\epsilon$ . Pick $x$ in the good set $G$ , since $(S^{1},R_{\alpha})$ is minimal, $o(x,R_{\alpha})$ has the given property. Since $\zeta_{k}$ restricted to $G$ is a homeomorphism and $\zeta_{k}R_{\alpha}=\sigma_{k}\zeta_{k}$ , $o(\zeta_{k}(x),R_{\alpha})$ has the desired property and further, $o(\zeta_{k}(x),R_{\alpha})$ is dense in $\zeta_{k}(G)$ and thus in $B_{k}(\alpha,\vec{\nu})=\operatorname{Cl}(\zeta_{k}(G))$ . Thus $B_{k}(\alpha,\vec{\nu})$ is minimal under $\sigma_{k}$ .

For part (b) note first that since $R_{p/q}$ is finite order and there are finitely many address intervals, $B_{k}(p/q,\vec{\nu})$ must consist of finitely many periodic orbits. The other properties in (a) and (b) will follow from (c) (proved using just these two partial results on (a) and (b)) and Theorem 7.9 using Remark 7.11(a).

For part (c), we first show that $B_{k}(\omega,\vec{\nu})$ is a recurrent symbolic kfsm set. By parts (a) and (b) we know that $B_{k}(\omega,\vec{\nu})$ is recurrent and by Remark 8.4 that $B_{k}(\omega,\vec{\nu})\subset\Omega_{k}$ and $\rho_{k}(B_{k}(\omega,\vec{\nu})=\omega$ . We show that $B_{k}(\omega,\vec{\nu})$ is symbolic kfsm set by showing that its full lift $A_{k}(\omega,\vec{\nu})$ to $\Omega_{\infty}$ is as required by Definition 8.1. As noted in Remark 2.2, ${\hat{T}}^{k}_{\infty}(A_{k}(\omega,\vec{\nu}))={\mathbb{A}}_{k}(\omega,\vec{\nu})$ so we need to show that ${\hat{\sigma}}_{\infty}$ is semi-monotone on $A_{k}(\omega,\vec{\nu})$ .

The first step is to show that $\zeta_{\infty}$ is weakly order preserving. Assume $x_{1}^{\prime},x_{2}^{\prime}\in G$ with $x_{1}^{\prime}<x_{2}^{\prime}$ . It could happen (when $\omega$ is rational) that $\zeta_{\infty}(x_{1}^{\prime})=\zeta_{\infty}(x_{2}^{\prime})$ , but if there exists a least $n$ with $(\zeta_{\infty}(x_{1}^{\prime}))_{n}\not=(\zeta_{\infty}(x_{2}^{\prime}))_{n}$ , then since $I_{m}<I_{m+1}$ for all $m$ and $R_{\omega}$ is order preserving, certainly $(\zeta_{\infty}(x_{1}^{\prime}))_{n}<(\zeta_{\infty}(x_{2}^{\prime}))_{n}$ , and so $\zeta_{\infty}$ is weakly order preserving.

We now show that ${\hat{\sigma}}_{\infty}$ is semi-monotone on $A_{k}(\omega,\vec{\nu})$ . Let $G$ be the good set for $\zeta_{\infty}$ and assume ${\underline{s}},{\underline{t}}\in\zeta_{\infty}(G)$ with ${\underline{s}}<{\underline{t}}$ . Then there exist $x_{1}^{\prime},x_{2}^{\prime}\in G$ with $\zeta_{\infty}(x_{1}^{\prime})={\underline{s}}$ and $\zeta_{\infty}(x_{2}^{\prime})={\underline{t}}$ and of necessity, $x_{1}^{\prime}<x_{2}^{\prime}$ and so $R_{\omega}(x_{1}^{\prime})<R_{\omega}(x_{2}^{\prime})$ . Since $\zeta_{\infty}R_{\omega}=\sigma_{\infty}\zeta_{\infty}$ , we have

\sigma_{\infty}({\underline{s}})=\sigma_{\infty}\zeta_{\infty}(x_{1}^{\prime})=\zeta_{\infty}R_{\omega}(x_{1})\leq\zeta_{\infty}R_{\omega}(x_{2})=\sigma_{\infty}\zeta_{\infty}(x_{2}^{\prime})=\sigma_{\infty}({\underline{t}}).

Thus $\sigma_{\infty}$ is weakly order preserving on $\zeta_{\infty}(G)$ and so on $A_{k}(\omega,\vec{\nu})$ . We have that $A_{k}(\omega,\vec{\nu})$ satisfies all the conditions of the lift in Definition 7.5 and thus $B_{k}(\omega,\vec{\nu})$ is symbolic kfsm.

Now for the converse assume that ${\hat{Z}}\subset\Omega_{k}$ is symbolic recurrent kfsm with ${\hat{\rho}}({\hat{Z}})=\omega$ . Let ${\hat{Z}}^{\prime}\subset\Omega_{\infty}$ be the lift that satisfies Definition 7.5. The proof splits into the two cases when $\omega$ is rational and irrational.

First assume $\omega=p/q$ with $\gcd(p,q)=1$ . We know from Lemma 7.9 and Remark 7.11 that ${\hat{Z}}$ consists of at most $k$ distinct periodic orbits each with period $kq/d$ with $d=\gcd(p,k)$ . We assume for simplicity that ${\hat{Z}}$ is a single periodic orbit. The case of multiple periodic orbits is similar but with more elaborate indexing.

For $i=0,\dots,kq/d-1$ let $P_{i}=(2i+1)d/2q\subset S_{k}$ and ${\mathcal{P}}=\{P_{i}\}$ . Since ${\hat{Z}}$ is a kfsm periodic orbit with $\rho_{k}$ -rotation number $p/q$ we may find an order preserving bijection $\phi:{\hat{Z}}\rightarrow{\mathcal{P}}$ with $\phi\sigma_{k}=R_{p/q}\phi$ on ${\hat{Z}}$ . Thus $\phi\sigma_{k}\phi^{-1}$ acts on ${\mathcal{P}}$ as $P_{i}\mapsto P_{i+p/d}$ reducing indices mod $kq/d$ .

For $j=0,\dots,k-1$ , let $X^{\prime}_{j}=\phi({\hat{Z}}\cap[j])$ where recall that $[j]$ is the length one cylinder set in $\Sigma^{+}_{2k}$ . Since $\phi$ is order preserving, each $X_{j}^{\prime}$ consists of a collection of adjacent points from ${\mathcal{P}}$ . If $X_{j}^{\prime}=\{P_{n(j)},\dots,P_{m(j)}\}\not=\emptyset$ , let $X_{j}=[P_{n(j)}-d/(2q),P_{m(j)}+d/(2q)]$ and when $X_{j}^{\prime}=\emptyset$ let $X_{j}=\emptyset$ . We now claim that $\{X_{j}\}$ is an address system as used in the HM-construction where $\vec{\nu}$ is defined by $\nu_{j+1}=|X_{2j}|$ and that $|X_{2j+1}|=p/q$ for $j=0,\dots,k-1$ yielding ${\hat{Z}}^{\prime}=B_{k}(p/q,\vec{\nu})$ .

Letting $\zeta$ be the itinerary map for the address system $\{X_{j}\}$ we have by construction that for ${\underline{s}}\in{\hat{Z}}$ that $\zeta\phi({\underline{s}})={\underline{s}}$ . In addition, for all $x\in[\phi({\underline{s}})-d/(2q),\phi({\underline{s}})+d/(2q)]$ we also have $\zeta(x)=\zeta\phi({\underline{s}})={\underline{s}}$ . Thus for any point $x$ in the good set $G$ , $\zeta(x)={\underline{s}}$ for some ${\underline{s}}\in{\hat{Z}}$ . This shows that ${\hat{Z}}=\operatorname{Cl}(\zeta(G))$ . The last step needed to show that ${\hat{Z}}=B_{k}(p/q,\vec{\nu})$ is to check that the address system is of the type used in the HM construction.

We need only check that $|X_{2j+1}|=p/q$ and for this it suffices to show that $\#X_{2j+1}=p/d$ . Assume first that $\#X_{2j+1}<p/d$ . Recalling that $\phi\sigma_{k}\phi^{-1}$ acts on the $X_{i}^{\prime}$ like $i\mapsto i+p/d$ , we see that there will be some $P_{m}\in X_{2j}^{\prime}$ and $P_{m+p/d}\in X_{2j+2}^{\prime}$ . Thus using $\phi^{-1}$ there is a ${\underline{s}}\in{\hat{Z}}$ with $s_{0}=2j$ and $s_{1}=2j+2$ , a contradiction to the fact that ${\hat{Z}}\subset\Omega_{k}$ and thus its allowable transitions are given by (5.2). On the other hand, if $\#X_{2j+1}>p/d$ we have some $P_{m}\in X_{2j+1}^{\prime}$ and $P_{m+p}\in X_{2j+1}^{\prime}$ again yielding a contradiction to ${\hat{Z}}\subset\Omega_{k}$ .

The irrational case is basically a continuous version of the rational one. By Remark 7.11(b) we have a continuous, onto $\phi:{\hat{Z}}\rightarrow S_{k}$ which is weakly order preserving, $\phi\sigma_{\infty}=R_{\alpha}\phi$ , and $\#\phi^{-1}(x)=1$ for all but a countable number of $R_{\alpha}$ -orbits on which $\#\phi^{-1}(x)=2$ .

For $j=0,\dots,k-1$ , let $X_{j}=\phi([j])$ . Thus $X_{j}$ is a closed interval (perhaps empty) with $\cup X_{j}={\mathbb{R}}$ , $X_{j}\leq X_{j+1}$ and adjacent intervals intersect only in their single common boundary point. We use $\{X_{j}\}$ as an address system with dynamics $R_{\alpha}$ , good set $G$ , and itinerary map $\zeta$ . By construction if ${\underline{s}}\in{\hat{Z}}$ with $\phi({\underline{s}})\in G$ , then ${\underline{s}}=\zeta\phi({\underline{s}})$ and so $\phi^{-1}(G)=\zeta\phi(\phi^{-1}(G))=\zeta(G)$ . Since ${\hat{Z}}$ is a Cantor set and $\phi^{-1}(G)$ is ${\hat{Z}}$ minus a countable set of $\sigma_{k}$ -orbits we have that $\phi^{-1}(G)$ is dense ${\hat{Z}}$ . Thus taking closures, ${\hat{Z}}=\operatorname{Cl}(\zeta(G))$ .

To finish we must show that $\{X_{j}\}$ is the type of address system allowable in the HM-construction. We just need $|X_{2j+1}|=\alpha$ for all $j$ . The proof is similar to the rational case. If $|X_{2j+1}|<\alpha$ then ${\hat{Z}}$ has a transition $2j\rightarrow 2j+2$ and if $|X_{2j+1}|>\alpha$ then ${\hat{Z}}$ has a transition $2j+1\rightarrow 2j+1$ . Either is a contradiction to ${\hat{Z}}\subset\Omega_{k}$ . Thus letting $\nu_{j+1}=|X_{2j}|$ for $j=0,\dots,k-1$ , we have ${\hat{Z}}=\operatorname{Cl}(\zeta(G))=B_{k}(\alpha,\vec{\nu})$ .

The last sentence in (c) follows from the construction of $\lambda_{k}(\omega,\vec{\nu})$ . ∎

Remark 8.6.

In Section 9.3 below we shall see that for the irrational case $\rho({\hat{Z}})=\omega\not\in{\mathbb{Q}}$ that there is a unique $\vec{\nu}$ with ${\hat{Z}}=B_{k}(\omega,\vec{\nu})$ and for rational $p/q$ that there are, in general, many $\vec{\nu}$ with $Z=B_{k}(p/q,\vec{\nu})$ . But note that if ${\hat{Z}}$ is a single periodic orbit then the proof above produces what we show is the unique $\vec{\nu}$ with ${\hat{Z}}=B_{k}(p/q,\vec{\nu})$ .

9. Parameterization of ${\mathcal{B}}_{k}(g)$ and ${\mathcal{N}}_{k}(g)$ by the HM construction

We know from Theorem 8.5(c) that the HM construction yields a correspondence between sets $B_{k}(\omega,\vec{\nu})$ and symbolic kfsm set in $\Omega_{k}$ . In addition, for a map $g\in{\mathcal{G}}$ using Theorem 7.6 we get a bijection from kfsm sets in $\Lambda_{k}(g)$ to those in ${\hat{\Lambda}}_{k}(g)\subset\Omega_{k}$ . Thus the HM construction provides a parameterization of ${\mathcal{B}}_{k}(g)$ . In this section we examine this parameterization as well as that of ${\mathcal{N}}_{k}(g)$ in detail.

9.1. Resonance and holes

As commented on above, the collection of all kfsm sets is closed in the compact metric space consisting of all compact $g$ -invariant sets with the Hausdorff topology. Thus the collection of all kfsm sets is complete. We have restricted attention here to recurrent kfsm sets or ${\mathcal{B}}_{k}(g)$ . This is because the recurrent ones are the most dynamically interesting and carry the invariant measures, but also as shown in Theorem 8.5, they are what is parameterized by the HM construction. As a consequence our primary space of interest ${\mathcal{B}}_{k}(g)$ is not complete, but rather has holes at points to be specified. What happens roughly is that as one takes the Hasudorff limit of recurrent kfsm sets the resulting kfsm set has homoclinic points that are not recurrent and so the limit is not recurrent and thus not any $B_{k}(\omega,\vec{\nu})$ . This is a phenomenon well known in Aubry-Mather theory. Another point of view on these “holes” is given in Section 13.2 using the family of interpolated semi-monotone maps.

In the HM construction fix $0<k<\infty$ . For a given allowable $(\omega,\vec{\nu})$ recall the address intervals are $X_{j}=X_{j\vec{\nu}}$ for $j=0,\dots,2k-1$ . Define $\ell_{j}=\ell_{j\vec{\nu}}$ and $r_{j}=r_{j\vec{\nu}}$ by $[\ell_{j},r_{j}]:=X_{j}$ . Note that $r_{j+1}=\ell_{j}$ with indices reduced mod $2k$ .

Definition 9.1.

The pair $(\omega,\vec{\nu})$ is called resonant if for some $n>1$ and $j,j^{\prime}$ , $R_{\omega}^{n}(\ell_{j})=\ell_{j^{\prime}}$ . A pair that is not resonant is called nonresonant.

Remark 9.2.

Note that for a rational $\omega=p/q$ all $(p/q,\vec{\nu})$ are resonant as are all $(\omega,\vec{\nu})$ when some $\nu_{i}=0$ . Also, for all $(\omega,\vec{\nu})$ and $j$ ,

R_{\omega}(\ell_{2j-1})=\ell_{2j}

(9.1)

which is the reason $n$ is restricted to $n>1$ in the definition.

The next lemma locates the “holes” in the space of all symbolic kfsm sets and thus in any ${\hat{{\mathcal{B}}}}(g)$ .

Lemma 9.3.

(a)

Assume $(\alpha,\vec{\nu})$ with $\alpha\not\in{\mathbb{Q}}$ is resonant. There exists a sequence $\vec{\nu}^{(i)}\rightarrow\vec{\nu}$ and a nonrecurrent kfsm $Z$ with $B_{k}(\alpha,\vec{\nu}^{(i)})\rightarrow Z$ in the Hausdorff topology on all compact subsets of $\Sigma^{+}_{2k}$ .
(b)

Assume $(p/q,\vec{\nu})$ with $p/q\in{\mathbb{Q}}$ . There exists a sequence $\omega^{(i)}\rightarrow p/q$ and a nonrecurrent kfsm $Z$ with $B_{k}(\omega^{(i)},\vec{\nu})\rightarrow Z$ in the Hausdorff topology on all compact subsets of $\Sigma^{+}_{2k}$ .

Proof.

We suppress the dependence on $k$ to simplify notation. For (a), the resonance hypothesis implies that there are odd $a$ and $b$ with $R_{\alpha}^{n}(X_{a\vec{\nu}})=X_{b\vec{\nu}}$ for some $n>0$ where we may assume $a<b$ . Since $R_{\alpha}^{n}(r_{a\vec{\nu}})=r_{b\vec{\nu}}$ by shrinking some $\nu_{j}$ for $a<j<b$ we obtain $\vec{\nu}^{\;\prime}$ and $x<r_{a\vec{\nu}}$ and arbitrarily close to it with $x\in G_{\alpha\vec{\nu}^{\;\prime}}$ and $R_{\alpha}^{n}(x)\in X_{b+1,\vec{\nu}^{\;\prime}}$ . In this way we can obtain sequences $\vec{\nu}^{(i)}\rightarrow\vec{\nu}$ and $x_{i}\nearrow r_{a\vec{\nu}}$ with $x_{i}\in G_{\alpha,\vec{\nu}^{(i)}}$ and $R_{\alpha}^{n}(x_{i})\in X_{b+1,\vec{\nu}^{(i)}}$ . Thus

\zeta_{\alpha,\vec{\nu}^{(i)}}(x_{i})=.a\dots(b+1)\zeta_{\alpha,\vec{\nu}^{(i)}}(R_{\alpha}^{n+1}(x_{i})

To simplify matters, assume that $R_{\alpha}(r_{b\vec{\nu}})\in G_{\alpha,\vec{\nu}}$ ; more complicated resonances are similar. Since $R_{\alpha}^{n+1}(x_{i})\rightarrow R_{\alpha}^{n+1}(r_{a\vec{\nu}})=R_{\alpha}(r_{b\vec{\nu}})$ using Remark 8.4(b),

\zeta_{\alpha,\vec{\nu}^{(i)}}(x_{i})\rightarrow.a\dots(b+1)\zeta_{\alpha,\vec{\nu}}(R_{\alpha}(r_{b\vec{\nu}}):={\underline{s}}.

Passing to a subsequence if necessary, by the compactness of the collection of symbolic kfsm sets there is a kfsm $Z$ with $B_{k}(\alpha,\vec{\nu}^{(i)})\rightarrow Z$ in the Hausdorff topology and by its construction, ${\underline{s}}\in Z$ . But ${\underline{s}}$ can’t be recurrent since by the resonance any length $(n+1)$ block in $\zeta_{\alpha,\vec{\nu}}(R_{\alpha}(r_{b\vec{\nu}})$ must start with $a$ and end in $b$ .

The argument for (b) is similar, but now the perturbation must be in $\omega$ since if $\omega=p/q$ is fixed, $R_{\omega}^{n}(X_{a})=X_{a}$ when $n=qk/\gcd(p,k)$ for all $\vec{\nu}$ . Fix an $a$ and so $R_{p/q}^{n}(r_{a})=r_{a}$ . By increasing $\omega$ incrementally we may find sequences $\omega^{(i)}\searrow p/q$ and $x_{i}\nearrow r_{a}$ with $x_{i}\in G_{\omega^{(i)},\vec{\nu}}$ so that the initial length $(n+1)$ block of $\zeta_{\omega^{(i)}\vec{\nu}}(x_{i})$ is $a\dots a+1$ . Thus if $\zeta_{p/q\vec{\nu}}(r_{a}+\epsilon)=P^{\infty}$ for small $\epsilon$ then

\zeta_{\omega^{(i)},\vec{\nu}}(x_{i})\rightarrow.a\dots(a+1)P_{2}P_{3}\dots P_{n-1}P^{\infty}:={\underline{t}}

where $P=(a+1)P_{2}P_{3}\dots P_{n-1}$ . As in the proof of (a) passing to a subsequence if necessary, there is a kfsm $Z$ with $B_{k}(\omega^{(i)},\vec{\nu})\rightarrow Z$ in the Hausdorff topology and by its construction, ${\underline{t}}\in Z$ . But ${\underline{t}}$ can’t be recurrent since any length $(n+1)$ block in $P^{\infty}$ must start and end with $a$ . ∎

9.2. continuity and injectivity

In doing the HM construction the explicit dependence of $A_{k}$ and $B_{k}$ on the pair $(\omega,\vec{\nu})$ was included. However, the elements of the pair have the interdependence $\sum\nu_{i}=k(1-\omega)$ and so when we treat $A_{k}$ and $B_{k}$ as functions it is sometimes better to eliminate the interdependence and treat them as functions of $\vec{\nu}$ alone, but the two variable version will also continue to be useful. Thus we sometimes overload the function $A_{k}$ and write

A_{k}(\vec{\nu})=A_{k}(1-\sum\nu_{i}/k,\vec{\nu})

and similarly for $B_{k}$ and the measure valued map $\lambda_{k}$ . The collection of allowable parameters for each $k$ is then

{\mathcal{D}}_{k}=\{\vec{\nu}\in{\mathbb{R}}^{k}\colon\nu_{i}\geq 0,\sum_{i=1}^{k}\nu_{i}\leq k\}.

The set of HM parameters corresponding to symbolic kfsm sets for $g\in{\mathcal{G}}$ is defined as

\operatorname{HM}_{k}(g)=\{\vec{\nu}\in{\mathcal{D}}_{k}\colon B_{k}(\vec{\nu})\subset{\hat{{\mathcal{B}}}}_{k}(g)\}.

Remark 9.4.

By Theorem 8.5, $B_{k}:\operatorname{HM}(g)\rightarrow{\hat{{\mathcal{B}}}}_{k}(g)$ is surjective and so $\iota_{k}^{-1}B_{k}:\operatorname{HM}(g)\rightarrow{\mathcal{B}}_{k}(g)$ provides a parameterization of the positive slope kfsm recurrent sets of $g\in{\mathcal{G}}$ and $(\iota_{k}^{-1})_{*}\lambda_{k}:\operatorname{HM}(g)\rightarrow{\mathcal{N}}_{k}(g)$ their invariant measures.

Example: The model map For the model map $f_{m}$ , $\operatorname{HM}_{k}(f_{m})={\mathcal{D}}_{k}$ since ${\hat{\Lambda}}_{k}(f)=\Omega_{k}$ .

The first issue in understanding what the HM construction tells us about ${\mathcal{B}}_{k}(g)$ and ${\mathcal{N}}_{k}(g)$ is to understand the nature of the maps $B_{k}$ and $\lambda_{k}$ . As indicated by Lemma 9.3 in the behaviour of the set-valued maps there is an essential distinction between the resonance and nonresonance cases.

Theorem 9.5.

Assume $g\in{\mathcal{G}}$ , for each $k>0$ ,

(a)

The map $(\iota_{k})^{-1}\circ B_{k}:\operatorname{HM}_{k}(g)\rightarrow{\mathcal{B}}_{k}(g)$ is onto and further it is continuous at nonresonant values and discontinuous at resonant values.
(b)

The map $(\iota_{k})^{-1}_{*}\circ\lambda_{k}:\operatorname{HM}_{k}(g)\rightarrow{\mathcal{N}}_{k}(g)$ is a homeomorphism and thus $\operatorname{HM}_{k}(g)$ is compact.

Proof.

Since we know from Lemma 7.8 that $\iota_{k}$ and $(\iota_{k})_{*}$ are homemorphisms we only consider $B_{k}$ and $\lambda_{k}$ . While these are functions of $\vec{\nu}$ alone, for the proof it is clearer to resort to the two variable versions with the proviso that $\omega=1-\sum\nu_{i}/k$ . Note that we have already shown in Theorem 8.5 that $\lambda_{k}$ and $B_{k}$ are onto ${\hat{{\mathcal{N}}}}_{k}(g)$ and ${\hat{{\mathcal{B}}}}_{k}(g)$ . We will often need to include the explicit dependence of various objects on the variables, for example, $\ell_{j}(\omega,\vec{\nu})$ , and we often suppress the dependence on $k$ .

We prove (b) first. We first show $\lambda_{k}$ is continuous. For each $j=1,\dots,2k-1$ and $i\in{\mathbb{N}}$ , let $\ell_{j}^{(i)}(\omega,\vec{\nu})=R_{\omega}^{-i}(\ell_{j}(\omega,\vec{\nu}))$ . The first observation from the HM construction is that

|\ell_{j}^{(i)}(\omega,\vec{\nu})-\ell_{j}^{(i)}(\omega_{0},\vec{\nu}_{0})|\leq\|(\omega,\vec{\nu})-(\omega_{0},\vec{\nu}_{0})\|_{1}

(9.2)

For a length $N$ block $B=b_{0}\dots b_{N-1}$ in $\Omega_{k}$ , let

Y_{B}(\omega,\vec{\nu})=\bigcap_{i=0}^{N-1}R^{-i}_{\omega}(\operatorname{Int}(X_{b_{i}}(\vec{\nu})))

and so $x\in G_{\omega,\vec{\nu}}\cap Y_{B}(\omega,\vec{\nu})$ implies that $\zeta_{\omega\vec{\nu}}(x)$ begins with the block $B$ . Also by the HM construction, $\lambda_{k}(\omega,\vec{\nu})([B])=m(Y_{B}(\omega,\vec{\nu}))$ with $m$ Lebesgue measure on the circle.

Recall that the weak topology on $\Sigma_{2k}^{+}$ is generated by the metric

d(\mu,\mu^{\prime})=\sum_{i=1}^{\infty}\frac{|\mu([B_{i}])-\mu^{\prime}([B_{i}])|}{2^{i}}

where $\{B_{i}\}$ is some enumeration of the blocks in $\Sigma_{2k}^{+}$ . Since each $Y_{B}(\omega,\vec{\nu})$ is a (perhaps empty) interval with endpoints some $\ell_{i}^{(j)}(\omega,\vec{\nu})$ , (9.2) implies that

|m(Y_{B}(\omega,\vec{\nu}))-m(Y_{B}(\omega_{0},\vec{\nu}_{0}))|\leq 2\|(\omega,\vec{\nu})-(\omega_{0},\vec{\nu}_{0})\|_{1}.

Thus summing over blocks

d(\lambda_{k}(\omega,\vec{\nu}),\lambda_{k}(\omega_{0},\vec{\nu}_{0})|\leq 2\|(\omega,\vec{\nu})-(\omega_{0},\vec{\nu}_{0})\|_{1}.

so $\lambda_{k}$ is continuous.

Since by definition in the HM construction, $\lambda_{k}(\omega,\vec{\nu})([2j])=\nu_{j+1}$ , $\lambda_{k}$ is injective. Recall now that for the model map, $\operatorname{HM}(f_{m})={\mathcal{D}}_{k}$ which is compact. So $\lambda_{k}:\operatorname{HM}(f_{m})\rightarrow{\hat{{\mathcal{N}}}}_{k}(f_{m})$ is a homeomorphism with image the set of all measures on recurrent symbolic kfsm sets in $\Omega_{k}$ . Thus, since $\operatorname{HM}_{k}(g)\subset{\mathcal{D}}_{k}$ we have that $\lambda_{k}:\operatorname{HM}_{k}(g)\rightarrow{\hat{{\mathcal{N}}}}_{k}(g)$ is also a homeomorphism. The compactness of ${\hat{{\mathcal{N}}}}_{k}(g)$ was proved in Lemma 7.8.

The proof of (a) is based on the following claim: $B_{k}$ is continuous at $(\omega_{0},\vec{\nu}_{0})$ if and only if for all $N$ there exits $\delta>0$ so that $\|(\omega,\vec{\nu})-(\omega_{0},\vec{\nu}_{0})\|<\delta$ implies that for all blocks $B$ of length $\leq N$ we have $Y_{B}(\omega_{0},\vec{\nu}_{0})$ nonempty exactly when $Y_{B}(\omega,\vec{\nu})$ is nonempty.

To prove the claim, first note that continuity is equivalent to the following: given $\epsilon>0$ there exists $\delta>0$ so that $\|(\omega,\nu)-(\omega_{0},\nu_{0})\|<\delta$ implies that for each ${\underline{s}}\in\zeta_{\omega\vec{\nu}}(G_{\omega\nu})$ there is a ${\underline{t}}\in\zeta_{\omega_{0}\vec{\nu}_{0}}(G_{\omega_{0}\vec{\nu}_{0}})$ with $d({\underline{s}},{\underline{t}})<\epsilon/2$ and for each ${\underline{t}}\in\zeta_{\omega_{0}\nu_{0}}(G_{\omega_{0}\nu_{0}})$ there is a ${\underline{s}}\in\zeta_{\omega\vec{\nu}}(G_{\omega\vec{\nu}})$ with $d({\underline{s}},{\underline{t}})<\epsilon/2$ . This implies that $\operatorname{HD}(\zeta_{\omega\vec{\nu}}(G_{\omega\vec{\nu}}),\zeta_{\omega_{0}\nu_{0}}(G_{\omega_{0}\nu_{0}})(G_{\omega_{0}\vec{\nu}_{0}}))<\epsilon/2$ and thus $\operatorname{HD}(B_{k}(\omega,\vec{\nu}),B_{k}(\omega_{0},\vec{\nu}_{0}))<\epsilon$ . Since $d({\underline{s}},{\underline{t}})$ is small exactly when ${\underline{s}}$ and ${\underline{t}}$ agree in a long prefix block $B$ and $Y_{B}(\omega,\vec{\nu})=\zeta_{\omega\vec{\nu}}^{-1}([B])$ , the claim follows.

We show that $B_{k}$ satisfies the condition in the claim when $(\omega,\vec{\nu})$ is nonresonant. Given $N$ for $j=1,\dots,2k-1$ and $i=0,\dots,N$ consider again $\ell_{j}^{(i)}(\omega,\vec{\nu})=R_{\omega}^{-i}(\ell_{j}(\omega,\vec{\nu})$ . By the HM construction we have $\ell_{2m}^{(n+1)}(\omega,\vec{\nu})=\ell_{2m-1}^{(n)}(\omega,\vec{\nu})$ for all $n$ , $m$ and $(\omega,\vec{\nu})$ . By nonresonance at $(\omega_{0},\vec{\nu}_{0})$ , all the other $\ell_{j}^{(i)}(\omega_{0},\vec{\nu}_{0})$ are disjoint. Since by (9.2) each $\ell_{j}^{(i)}(\omega,\vec{\nu})$ depends continuously on $(\omega,\vec{\nu})$ and the endpoints of each $Y_{B}(\omega,\vec{\nu})$ is some $\ell_{j}^{(i)}(\omega,\vec{\nu})$ , we may find a $\delta$ so that $\|(\omega,\nu)-(\omega_{0},\nu_{0})\|<\delta$ implies that the $\ell_{j}^{(i)}(\omega,\vec{\nu})$ are ordered around $S_{k}$ in the same way and with the same gaps between them as the $\ell_{j}^{(i)}(\omega_{0},\vec{\nu}_{0})$ . This implies that for each block $B$ of length $B\leq N$ , $Y_{B}(\omega_{0},\vec{\nu}_{0})$ is nonempty exactly when $Y_{B}(\omega,\vec{\nu})$ is nonempty and so $B_{k}$ is continuous.

For the discontinuity, since the sets $Z$ in Lemma 9.3(ab) are not recurrent, they are not equal to $B_{k}(\omega_{0},\vec{\nu}_{0})$ .

∎

Remark 9.6.

(a)

The parameter space ${\mathcal{D}}_{k}$ is $(k-1)$ -dimensional. Assuming $\omega\not\in{\mathbb{Q}}$ , then for a fixed $n>1$ and $j,j^{\prime}$ , the collection of all $\vec{\nu}\in{\mathcal{D}}_{k}$ which yield $R_{\omega}^{n}(\ell_{j})=\ell_{j^{\prime}}$ is a $(k-2)$ -dimensional affine subspace. Thus the set of resonance parameters is a countable dense collection of codimension one affine subspaces and so the resonance case is full measure and dense $G_{\delta}$
(b)

One can show that $B_{k}$ is lower semicontinuous [10], in particular if $(\omega^{(i)},\vec{\nu}^{(i)})\rightarrow(\omega,\vec{\nu})$ and some subsequence of $B_{k}(\omega^{(i)},\vec{\nu}^{(i)})$ converges to $Z$ in the Hausdorff topology, then $B_{k}(\omega,\vec{\nu})\subset Z$ . The Semi-Continuity Lemma (see page 114 of [18]) yields that a lower semi-continuous set-valued function is continuous on a dense $G_{\delta}$ set. In the case of $(\iota_{k})^{-1}\circ B_{k}$ the last theorem exactly identifies this continuity set as the nonresonant $(\omega,\vec{\nu})$ .

9.3. Slices and skewness

Since $\rho$ and ${\hat{\rho}}$ are defined and continuous on the various spaces ${\mathcal{B}}_{k}(g)$ etc., we may define the closed slices with a given rotation number.

Definition 9.7.

For $g\in{\mathcal{G}}$ let ${\mathcal{B}}_{k\omega}(g)=\{Z\in{\mathcal{B}}_{k}(g)\colon\rho(Z)=\omega\}$ and ${\hat{{\mathcal{B}}}}_{k\omega}(g)=\{Z\in{\hat{{\mathcal{B}}}}_{k}(g)\colon{\hat{\rho}}(Z)=\omega\}$ and the restriction of $\iota_{k}$ to ${\mathcal{B}}_{k\omega}(g)$ is denoted $\iota_{k\omega}$ The slices of invariant measures ${\mathcal{N}}_{k\omega}(g)$ and ${\hat{{\mathcal{N}}}}_{k\omega}(g)$ are defined similarly. The $\omega$ -slice of HM parameters is $\operatorname{HM}_{k\omega}(g)=B_{k}^{-1}({\mathcal{B}}_{k\omega}(g))=\lambda_{k}^{-1}({\mathcal{N}}_{k\omega}(g))$

Definition 9.8.

For $p/q\in{\mathbb{Q}}$ an allowable parameter $\vec{\nu}$ is called pure if $B_{k}(p/q,\vec{\nu})$ consists of a single periodic orbit. The collection of $p/q$ pure parameters is denoted $\operatorname{Pure}_{k,p/q}\subset{\mathcal{D}}_{k,p/q}$ and it will be shown in Lemma 12.4 to be an affine lattice. For a $g\in{\mathcal{G}}$ its pure parameters are $\operatorname{Pure}_{k,p/q}(g)=\operatorname{HM}_{k,p/q}(g)\cap\operatorname{Pure}_{k,p/q}$ .

Remark 9.9.

For a given symbolic kfsm $p/q$ -periodic orbit $P$ by Theorem 8.5(c) there is some $\vec{\nu}$ with $B_{k}(p/q,\vec{\nu})=P$ . Since a periodic orbit is uniquely ergodic and $\lambda_{k}$ is injective this $\vec{\nu}$ is unique. Thus there is a bijection between symbolic kfsm $p/q$ -periodic orbits and $\operatorname{Pure}_{k,p/q}$ .

Lemma 9.10.

Assume $g\in{\mathcal{G}}$

(a)

For all $\omega$ , $(\iota_{k\omega}^{-1})_{*}\circ\lambda_{k\omega}:\operatorname{HM}_{k\omega}(g)\rightarrow{\mathcal{N}}_{k\omega}(g)$ is a homeomorphism.
(b)

When $\alpha\not\in{\mathbb{Q}}$ , $\iota_{k\alpha}^{-1}\circ B_{k\alpha}:\operatorname{HM}_{k\alpha}(g)\rightarrow{\mathcal{B}}_{k\alpha}(g)$ is injective as well as continuous at nonresonant $(\alpha,\vec{\nu})$ and discontinuous at resonant $(\alpha,\vec{\nu})$ .
(c)

When $p/q\in{\mathbb{Q}}$ , $\iota_{kp/q}^{-1}B_{kp/q}:\operatorname{HM}_{kp/q}(g)\rightarrow{\mathcal{B}}_{kp/q}(g)$ is injective on $\operatorname{Pure}_{k,p/q}$ .

Proof.

Since $\iota_{k}$ restricts to a homeomorphism on slices we only consider $B_{k\omega}$ and $\lambda_{k\omega}$ . Part (a) follows immediately from Theorem 9.5.

For (b), when $\alpha\not\in{\mathbb{Q}}$ the assignment of a semi-Denjoy kfsm set with rotation number $\alpha$ to its unique invariant measure yields a bijection ${\mathcal{B}}_{k\alpha}(g)\rightarrow{\mathcal{N}}_{k\alpha}(g)$ and ${\hat{{\mathcal{B}}}}_{k\alpha}(g)\rightarrow{\hat{{\mathcal{N}}}}_{k\alpha}(g)$ . Since by (a), $\lambda_{k\alpha}$ is injective, we have that $B_{k\alpha}$ is also. Continuity of $B_{k\alpha}$ at nonresonant values on irrational slices follows directly from (a). Discontinuity at resonant values on irrational slices follows from Lemma 9.3(a).

∎

Remark 9.11.

Since ${\mathcal{B}}_{kp/q}(g)$ is a finite, set the continuity of $\iota_{kp/q}^{-1}B_{kp/q}:\operatorname{HM}_{kp/q}(g)\rightarrow{\mathcal{B}}_{kp/q}(g)$ is not particularly interesting, but we will remark on it in Section 12.4.

The skewness $\gamma(\mu)$ of a $\tilde{g}_{k}$ -invariant measure in $S_{k}$ is the amount of measure in each fundamental domain. When its $j^{th}$ component is large, roughly its $\tilde{g}_{k}$ -orbits are moving slowly through $[j-1,j)$ . When we project to the base $S^{1}$ in the next section the skewness will thus indicate how quickly orbits are moving of the $j^{th}$ loop of the kfsm set.

Definition 9.12.

Assume $g\in{\mathcal{G}}$

(a)

For $\eta\in{\mathcal{N}}_{k}(g)$ , $\gamma(\eta)=(\eta([0,1)),\eta([1,2)),\dots,\eta([k-1,k))$
(b)

For ${\hat{\eta}}\in{\hat{{\mathcal{N}}}}_{k}(g)$ , ${\hat{\gamma}}(\eta)=({\hat{\eta}}([0]\cup[1]),{\hat{\eta}}([2]\cup[3])\dots,{\hat{\eta}}([2k-2]\cup[2k-1])$

Note that the skewness takes values in the unit simplex $\sum a_{i}=1$ , $a_{i}\geq 0$ and contains no information about the rotation number.

Lemma 9.13.

Assume $g\in{\mathcal{G}}$

(a)

${\hat{\gamma}}\circ(\iota_{k})_{*}=\gamma$
(b)

$\gamma(\lambda_{k}(\omega,\vec{\nu}))=(\omega+\nu_{1},\omega+\nu_{2},\dots,\omega+\nu_{k})/k$
(c)

For $\eta\in{\mathcal{N}}_{k\omega}$ , $\gamma_{1}(\eta)=k\gamma(\eta)-\omega\mathbbm{1}$ is inverse to $(\iota_{k})_{*}^{-1}\circ\lambda_{k}$ and so it is a homeomorphism.
(d)

$\gamma$ is a homeomorphism from ${\mathcal{N}}_{k\omega}(g)$ onto its image as is ${\hat{\gamma}}$ from ${\hat{{\mathcal{N}}}}_{k\omega}(g)$ onto its image

Remark 9.14.

The last lemma formalizes the description in the Introduction on the
parametrization of the weak disks of semi-Denjoy minimal sets by their speed in each “loop” around the circle. For rational pure parameters the skewness counts the number of elements in each fundamental domain and this thus yields a discrete parametrization of the kfsm $p/q$ -periodic orbits.

10. kfsm sets in $S^{1}$ and $\Omega_{1}$

10.1. In $S^{1}$

We now return to our central concern, $g$ -invariant sets in $S^{1}$ that have a lift to $S_{k}$ that is semi-monotone. Once again the definition makes sense for any degree one circle map but we restrict to the class ${\mathcal{G}}$ .

Definition 10.1.

Given $g\in{\mathcal{G}}$ , a compact $g$ -invariant set $Z\subset S^{1}$ is kfsm if it has a $\tilde{g}$ -invariant lift $Z^{\prime}\subset{\mathbb{R}}$ which is kfsm, or equivalently, $Z$ has a $\tilde{g}_{k}$ -invariant lift $Z^{*}\subset S_{k}$ which is kfsm. Let ${\mathcal{C}}_{k}(g)$ be all compact, invariant, recurrent kfsm sets in $\Lambda_{1}(g)$ with the Hausdorff topology and ${\mathcal{O}}_{k}(g)$ be all $g$ -invariant, Borel probability measures supported on $Z\in{\mathcal{C}}_{k}(g)$ with the weak topology

Thus when $Z$ is kfsm, it has a lift to $S_{k}$ which is semi-monotone under the action of $\tilde{g}_{k}$ on its lift.

To make contact with the usual definitions in Aubry-Mather theory, assume that $x\in S^{1}$ is such that $o(x,f)$ is k-fold semi-monotone. This happens exactly when there is a point $x^{\prime}\in{\mathbb{R}}$ with $\pi_{\infty}(x^{\prime})=x$ and for all positive integers $\ell,m,n$ ,

\tilde{g}^{\ell}(x^{\prime})<T^{km}\tilde{g}^{n}(x^{\prime})\ \ \text{implies}\ \ \tilde{g}^{\ell+1}(x^{\prime})\leq T^{m}\tilde{g}^{n+1}(x^{\prime})

In Aubry-Mather theory one would write $x_{j}=\tilde{g}^{j}(x^{\prime})$ .

Remark 10.2.

(a)

$\pi_{k}:S_{k}\rightarrow S^{1}$ induces continuous onto maps ${\mathcal{B}}_{k}(g)\rightarrow{\mathcal{C}}_{k}(g)$ and ${\mathcal{N}}_{k}(g)\rightarrow{\mathcal{O}}_{k}(g)$ .
(b)

$Z^{*}\subset S_{k}$ is kfsm if and only if $\pi_{k}(Z^{*})\subset S^{1}$ is.
(c)

If $Z\subset S^{1}$ is is a $k$ -fold semi-monotone then it is also $\ell k$ -fold semi-monotone for any $\ell>0$ .
(d)

If $P$ is a periodic orbit of $g$ of type $(p,q)$ (which are perhaps not relatively prime) then $P$ has a lift $P^{\prime}$ to ${\mathbb{R}}$ with $T^{p}(P^{\prime})=P^{\prime}$ and is monotone since $g\in{\mathcal{G}}$ implies $\tilde{g}(x^{\prime})\geq x^{\prime}$ and so $P$ is automatically $p$ -fsm.
(e)

Using Lemma 7.6 a recurrent kfsm set in $S^{1}$ is either a collection of periodic orbits all with the same rotation number (a cluster) or else a semi-Denjoy minimal set. A minimal kfsm set $S^{1}$ is either a single periodic orbit or else a semi-Denjoy minimal set.
(f)

A collection of periodic orbits all with the same rotation number that individually are kfsm when considered as a set is not of necessity a kfsm (i.e., a cluster)

10.2. Symbolic kfsm sets in $\Omega_{1}$

We now consider symbolic kfsm sets in the symbolic base $\Omega_{1}=\Sigma_{2}^{+}$ .

Definition 10.3.

A $\sigma_{1}$ -invariant set ${\hat{Z}}\subset\Omega_{1}=\Sigma_{2}$ is kfsm if there is a $\sigma_{\infty}$ -invariant lift ${\hat{Z}}^{\prime}$ (i.e., ${\hat{p}}_{\infty}({\hat{Z}}^{\prime})={\hat{Z}}$ ) which is kfsm or equivalently, ${\hat{Z}}$ has a $\sigma_{k}$ -invariant lift ${\hat{Z}}^{*}\subset\Omega_{k}$ which is kfsm. Given $g\in{\mathcal{G}}$ let ${\hat{{\mathcal{C}}}}_{k}(g)$ be all recurrent kfsm sets in ${\hat{\Lambda}}_{1}(g)$ with the Hausdorff topology and ${\hat{{\mathcal{O}}}}_{k}(g)$ be all $g$ -invariant, Borel probability measures supported on ${\hat{Z}}\in{\hat{{\mathcal{C}}}}_{k}(g)$ with the weak topology

Using Theorem 7.6 we connect kfsm sets in $\Lambda_{1}(g)$ to their symbolic analogs in ${\hat{\Lambda}}_{1}(g)$ and get

Corollary 10.4.

A $g$ invariant set $Z\subset\Lambda_{1}(g)$ is kfsm if an only if $\iota_{1}(Z)\subset{\hat{\Lambda}}_{1}(g)$ is. Further, $\iota_{1}$ induces homeomorphisms ${\mathcal{C}}_{k}(g)\rightarrow{\hat{{\mathcal{C}}}}_{k}(g)$ and ${\mathcal{O}}_{k}(g)\rightarrow{\hat{{\mathcal{O}}}}_{k}(g)$ .

Remark 10.5.

All the comments in Remark 10.2 hold for symbolic kfsm sets mutatis mutandis.

10.3. The HM construction and its symmetries

We bring the HM construction back into play and take the projections from $\Omega_{k}$ to $\Omega_{1}$ .

Definition 10.6.

Let $C_{k}(\omega,\vec{\nu})={\hat{\pi}}_{k}(B_{k}(\omega,\vec{\nu}))$ and $\mu_{k}(\omega,\vec{\nu})=({\hat{\pi}}_{k})_{*}(\lambda_{k}(\omega,\vec{\nu}))$

We know from Theorem 9.5 that the HM construction provides a parameterization of ${\mathcal{B}}_{k}(g)$ and ${\mathcal{N}}_{k}(g)$ , the goal now is to get a parameterization of the kfsm sets and their invariant measures in $S^{1}$ , i.e., of ${\hat{{\mathcal{C}}}}_{k}(g)$ and ${\hat{{\mathcal{O}}}}_{k}(g)$ . For this we need to understand the symmetries inherent in the HM construction.

Recall the left shift on the parameter $\nu$ is $\tau(\nu_{1},\dots,\nu_{k})=(\nu_{2},\dots,\nu_{k},\nu_{1})$ . There are two types of symmetries to be considered. The first is when different $\vec{\nu}$ give rise to the same $C_{k}(\omega,\vec{\nu})$ . For minimal $C_{k}(\omega,\vec{\nu})$ this happens if and only if the $\vec{\nu}$ ’s are shifts of each other as is stated in parts (a) and (d) in the next theorem. The second sort of symmetry happens when some $C_{k}(\omega,\vec{\nu})$ is also a $C_{j}(\omega,\vec{\nu}^{\;\prime})$ for some $j<k$ , which is to say the map ${\hat{\pi}}_{k}:B_{k}(\omega,\vec{\nu})\rightarrow C_{k}(\omega,\vec{\nu})$ is not one-to-one. In the minimal case this happens if and only if $\tau^{j}(\vec{\nu})=\vec{\nu}$ as is stated in parts (b) and (c) below.

Lemma 10.7.

Fix $k>0$ and assume $\vec{\nu}$ is allowable for $\omega$ .
(a)

For all $j$ , $B_{k}(\omega,\tau^{j}(\vec{\nu}))={\hat{T}}_{k}^{j}(B_{k}(\omega,\nu))$ and so $C_{k}(\omega,\tau^{j}(\vec{\nu}))=C_{k}(\omega,\nu)$
(b)

If $\tau^{j}(\vec{\nu})=\vec{\nu}$ for some $0<j<k$ then

$B_{k}(\omega,\vec{\nu})={\hat{T}}_{k}^{j}(B_{k}(\omega,\vec{\nu}))$ (10.1)

and $C_{k}(\omega,\vec{\nu})=C_{j}(\omega,\vec{\nu}^{\;\prime})$ where $\vec{\nu}^{\;\prime}=(\nu_{1},\dots,\nu_{j})$ .
(c)

If $B_{k}(\omega,\vec{\nu})$ is minimal and (10.1) holds then $\vec{\nu}=\tau^{j}(\vec{\nu})$ . If $\vec{\nu}\not=\tau^{j}(\vec{\nu})$ for all $0<j<k$ , then ${\hat{\pi}}_{k}:B_{k}(\omega,\vec{\nu})\rightarrow C_{k}(\omega,\vec{\nu})$ is a homeomorphism.
(d)

If $B_{k}(\omega,\vec{\nu})$ and $B_{k}(\omega,\vec{\nu}^{\;\prime})$ are minimal and $C_{k}(\omega,\vec{\nu})=C_{k}(\omega,\vec{\nu}^{\;\prime})$ , then for some $j$ , $\vec{\nu}^{\;\prime}=\tau^{j}(\vec{\nu})$ .

Proof.

The fact that $B_{k}(\omega,\tau^{j}(\vec{\nu}))={\hat{T}}_{k}^{j}(B_{k}(\omega,\nu))$ is an easy consequence of the HM construction and since ${\hat{\pi}}_{k}{\hat{T}}^{k}={\hat{\pi}}_{k}$ we have $C_{k}(\omega,\tau^{j}(\vec{\nu}))=C_{k}(\omega,\nu)$ , proving (a) The first part of (b) follows directly from (a) using the given fact that $\tau^{j}(\vec{\nu})=\vec{\nu}$ .

For the second part of (b), first note that if $\{X_{i}\}$ is the address system for $k$ and $(\omega,\vec{\nu})$ then since $\tau^{j}(\vec{\nu})=\vec{\nu}$ , we have $T_{k}^{j}(X_{i})=X_{i+2j}$ . This implies that under the quotient $S_{k}\rightarrow S_{j}$ , $\{X_{i}\}$ descends to an allowable HM-address system on $S_{j}$ using $(\omega,\vec{\nu}^{\;\prime})$ . Thus using the dynamics $R_{\omega}$ on both address systems, the corresponding entries of $B_{k}(\omega,\vec{\nu})$ and $B_{j}(\omega,\vec{\nu}^{\;\prime})$ are equal $\mod 2$ and so $C_{k}(\omega,\vec{\nu})=C_{j}(\omega,\vec{\nu}^{\;\prime})$ .

To prove the first part of (c), as remarked in Remark 7.11, if $B_{k}(\omega,\nu)$ is minimal it is uniquely ergodic. Thus if (10.1) holds, then $\lambda_{k}(\omega,\vec{\nu})=\lambda_{k}(\omega,\tau^{j}(\vec{\nu}))$ and since $\lambda_{k}$ is injective by Theorem 9.5, $\vec{\nu}=\tau^{j}(\vec{\nu})$ . Now for the second part of (c), certainly ${\hat{\pi}}_{k}:B_{k}(\omega,\vec{\nu})\rightarrow C_{k}(\omega,\vec{\nu})$ is continuous and onto, so assume it is not injective. Then there exists ${\underline{s}},{\underline{t}}\in B_{k}(\omega,\vec{\nu})$ with ${\underline{s}}\not={\underline{t}}$ and ${\hat{\pi}}_{k}({\underline{s}})={\hat{\pi}}_{k}({\underline{t}})$ . Thus for some $0<j^{\prime}<k$ , ${\underline{t}}={\hat{T}}_{k}^{j^{\prime}}({\underline{s}})$ and so if $j=k-j^{\prime}$ , $B_{k}(\omega,\vec{\nu})\cap T_{k}^{j}B_{k}(\omega,\vec{\nu})\not=\emptyset$ . But by assumption $B_{k}(\omega,\vec{\nu})$ is minimal and so $B_{k}(\omega,\vec{\nu})=T_{k}^{j}B_{k}(\omega,\vec{\nu})$ and so $\vec{\nu}=\tau^{j}(\vec{\nu})$ , a contradiction. Thus ${\hat{\pi}}_{k}:B_{k}(\omega,\vec{\nu})\rightarrow C_{k}(\omega,\vec{\nu})$ is injective, as required

For part (d), $C_{k}(\omega,\vec{\nu})=C_{k}(\omega,\vec{\nu}^{\;\prime})$ implies that

\cup_{i=1}^{k}T^{i}(B_{k}(\omega,\vec{\nu}))={\hat{\pi}}^{-1}(C_{k}(\omega,\vec{\nu}))={\hat{\pi}}^{-1}(C_{k}(\omega,\vec{\nu}^{\;\prime}))=\cup_{i=1}^{k}T^{i}(B_{k}(\omega,\vec{\nu}^{\;\prime})).

Since each $T^{i}(B_{k}(\omega,\vec{\nu}))$ and $T^{i}(B_{k}(\omega,\vec{\nu}^{\;\prime}))$ is minimal, for some $j$ , $T^{j}(B_{k}(\omega,\vec{\nu}))=B_{k}(\omega,\vec{\nu}^{\;\prime})$ , and so by part (c), $\tau^{j}(\vec{\nu})=\vec{\nu}^{\;\prime}$ . ∎

Remark 10.8.

It is possible that if $B_{k}(p/q,\vec{\nu})$ is a cluster of periodic orbits, $\pi_{k}$ could be injective on some of them and not on others.

10.4. continuity and injectivity

Let $\overline{\operatorname{HM}_{k}(g)}=\operatorname{HM}_{k}(g)/\tau$ with equivalence classes denoted $[\vec{\nu}]$ . Note that $\tau^{j}(\vec{\nu})\in{\mathcal{D}}_{k}$ for some $j$ is resonant if and only if $\vec{\nu}$ is, so we may call $[\vec{\nu}]$ resonant or nonresonant.

Since the $\tau$ action preserves slices we define $\overline{\operatorname{HM}}_{k\omega}=\operatorname{HM}_{k\omega}(g)/\tau$ . The $\omega$ -slices of ${\mathcal{C}}_{k}(g)$ and ${\mathcal{O}}_{k}(g)$ are defined in the obvious way. If $(p/q,\vec{\nu})$ is a pure parameter so is $\tau^{j}(\vec{\nu})$ for any $j$ and so we define $\overline{\operatorname{Pure}}(k,p/q)=\operatorname{Pure}(k,p/q)/\tau$ . Note that $\overline{\operatorname{Pure}}(k,p/q)$ is all $[\vec{\nu}]$ such that $B_{k}(p/q,\vec{\nu})$ is a single periodic orbit it is not all $[\vec{\nu}]$ such that $C_{k}(p/q,\vec{\nu})={\hat{\pi}}_{k}B_{k}(p/q,\vec{\nu})$ is a single periodic orbit.

Definition 10.9.

Lemma 10.7 implies that $(\iota_{1})^{-1}\circ C_{k}$ induces a map $\theta_{k}:\overline{\operatorname{HM}}_{k}(g)\rightarrow{\mathcal{C}}_{k}(g)$ and that $(\iota_{1})^{-1}_{*}\circ\mu_{k}$ induces a map $\beta_{k}:\overline{\operatorname{HM}}_{k}(g)\rightarrow{\mathcal{O}}_{k}(g)$ . The induced maps on slices are $\theta_{k\omega}:\overline{\operatorname{HM}}_{k\omega}(g)\rightarrow{\mathcal{C}}_{k\omega}(g)$ and $\beta_{k\omega}:\overline{\operatorname{HM}}_{k\omega}(g)\rightarrow{\mathcal{O}}_{k\omega}(g)$

Theorem 10.10.

Assume $g\in{\mathcal{G}}$ , for each $k>0$ ,

(a)

The map $\theta_{k}$ is onto, continuous at nonresonant values and discontinuous at resonant values. Restricted to an irrational slice it is injective, continuous at nonresonant values and discontinuous at resonant values. Restricted to a rational slices it is injective on the pure lattice.
(b)

The map $\beta_{k}$ is a homeomorphism when restricted to irrational slices and pure rational lattices.

Proof.

By construction we have the following commuting diagram

\begin{CD}\operatorname{HM}_{k}(g)@>{B_{k}}>{}>{\hat{{\mathcal{B}}}}_{k}(g)@>{\iota_{k}^{-1}}>{}>{\mathcal{B}}_{k}(g)\\ @V{\sim}V{}V@V{}V{{\hat{\pi}}_{k}}V@V{}V{\pi_{k}}V\\ \overline{\operatorname{HM}}_{k}(g)@>{C_{k}}>{}>{\hat{{\mathcal{C}}}}_{k}(g)@>{\iota_{1}^{-1}}>{}>{\mathcal{C}}_{k}(g)\end{CD}

(10.2)

with the vertical maps all onto and continuous and $\theta_{k}$ the composition of the bottom horizontal maps and the map $C_{k}$ also denotes the map induced on equivalence classes in $\overline{\operatorname{HM}}_{k}(g)$ . Since the given versions of ${\iota_{k}}$ and $\iota_{1}$ are homeomorphisms we need only consider $C_{k}$ and $\mu_{k}$ . The fact that these are continuous follows from Lemma 9.10 and the just stated properties of the diagram as do the various continuity assertions in the theorem. We prove the discontinuity result for $C_{k}$ on irrational slices. The other discontinuity assertions follow similarly.

Assume $(\alpha,\vec{\nu})$ is resonant with $\alpha\not\in{\mathbb{Q}}$ . From Lemma 9.3 and its proof we have a sequence $(\alpha,\vec{\nu}^{(i)})\rightarrow(\alpha,\vec{\nu})$ so that $B_{k}(\alpha,\vec{\nu}^{(i)})\rightarrow Z$ and a ${\underline{s}}\in Z\setminus B_{k}(\alpha,\vec{\nu})$ with ${\underline{s}}$ nonrecurrent. In the quotients $[\vec{\nu}^{(i)}]\rightarrow[\vec{\nu}]$ and $C_{k}(\alpha,\vec{\nu}^{(i)})\rightarrow{\hat{\pi}}_{k}(Z)$ by continuity. We need to show that ${\hat{\pi}}_{k}(Z)\not=C_{k}(\alpha,\vec{\nu})$ Now if $\pi_{k}({\underline{s}})\in{\hat{\pi}}_{k}(Z)\setminus C_{k}(\alpha,\vec{\nu})$ we are done so assume $\pi_{k}({\underline{s}})\in C_{k}(\alpha,\vec{\nu})$ . Thus for some ${\underline{t}}\in B_{k}(\alpha,\vec{\nu})$ , $\pi_{k}({\underline{s}})=\pi_{k}({\underline{t}})$ and so by Lemma 5.3(e), for some $j$ , ${\hat{T}}^{j}_{k}({\underline{s}})={\underline{t}}$ . This implies that the action of $\sigma_{k}$ on $\operatorname{Cl}(o({\underline{s}},\sigma_{k}))$ is conjugated to that on $\operatorname{Cl}(o({\underline{t}},\sigma_{k}))$ by ${\hat{T}}_{k}^{j}$ . But by the classification Theorem 8.5, $\operatorname{Cl}(o({\underline{t}},\sigma_{k}))$ is a minimal set and thus so is $\operatorname{Cl}(o({\underline{s}},\sigma_{k}))$ and so ${\underline{s}}$ is recurrent, a contradiction, yielding the discontinuity.

To show $C_{k}$ is injective on the sets indicated, assume $C_{k}(\omega,\vec{\nu})=C_{k}(\omega,\vec{\nu}^{\;\prime})$ with either $\omega=p/q$ and $\vec{\nu},\vec{\nu}^{\;\prime}$ in the pure lattice or $\omega\not\in{\mathbb{Q}}$ . In either case $B_{k}(\omega,\vec{\nu})$ and $B_{k}(\omega,\vec{\nu}^{\;\prime})$ are minimal and since ${\hat{\pi}}_{k}$ is a semiconjugacy, $C_{k}(\omega,\vec{\nu})$ and $C_{k}(\omega,\vec{\nu}^{\;\prime})$ are also. Thus by Lemma 10.7(d), for some $j$ , $\vec{\nu}=\tau^{j}\vec{\nu}^{\;\prime}$ and so $[\vec{\nu}]=[\vec{\nu}^{\;\prime}]$ .

Now for part (b), there is a diagram similar to (10.2) for $\beta_{k}$ . Since Denjoy minimal sets and individual periodic orbits are uniquely ergodic, the injectivity asserted for $\mu_{k}$ follows from that of $C_{k}$ just proved. Continuity and surjectivity follow from the diagram and Lemma 9.10. ∎

Remark 10.11.

We remark on the relationship of pure parameters to $C_{k}$ and $B_{k}$ . As a short hand we indicate symbolic periodic orbits by their repeating block. A simple computation shows that $B_{2}(2/5,(3/5,3/5))=01223\ \cup\ 00123$ and so $C_{2}(2/5,(3/5,3/5))=01001$ . Note that as required ${\hat{T}}_{2}(01223)=00123$ and $01001$ is the $2/5$ -Sturmian. Now $B_{2}(2/5,(4/5,2/5))=00123$ and so $C_{2}(2/5,(4/5,2/5))=01001=C_{2}(2/5,(3/5,3/5))$ . A further computation shows that both $\mu_{2}(2/5,(4/5,2/5))$ and $\mu_{2}(2/5,(3/5,3/5))$ are the unique invariant measure on $01001$ and thus $\mu_{2}$ is not injective on rational slices of $\overline{\operatorname{HM}}_{k}$ despite the fact that it is injective on rational slices of $\operatorname{HM}_{k}$ . The underlying explanation is that being a pure parameter requires $B_{k}$ to be a single periodic orbit not that $C_{k}$ be one.

Definition 10.12.

Let $\mathcal{Q}_{k}=P_{k}/\tau$ where $P_{k}\subset{\mathbb{R}}^{k+1}$ is the standard $k$ -dimensional simplex and $\tau$ is the shift. Equivalence classes in $\mathcal{Q}_{k}$ are denoted $[\cdot]$ . For ${\hat{\eta}}\in{\hat{{\mathcal{O}}}}_{k}(g)$ with $\rho({\hat{\eta}})=\omega$ from Theorem 10.10 we may find an $\vec{\nu}$ with ${\hat{\eta}}=\mu_{k}(\omega,\vec{\nu})$ . The skewness of ${\hat{\eta}}$ is defined as $\overline{\gamma}({\hat{\eta}}):=[\gamma(\lambda_{k}(\omega,\vec{\nu}))]$ . Note that by Lemma 10.7 this is independent of the choice of $\mu_{k}(\omega,\vec{\nu})$ . And also for $\eta\in{\mathcal{O}}_{k}(g)$ via $\overline{\gamma}(\eta)=\overline{\gamma}((\iota_{k})_{*}(\eta))$ .

Remark 10.13.

On an irrational quotient slice ${\mathcal{O}}_{k,\alpha}$ , let $\overline{\gamma}_{1}=k\overline{\gamma}-\alpha\mathbbm{1}$ , then $\overline{\gamma}_{1}$ is the inverse of $\beta_{k}$ and may be viewed as a parameterization of ${\hat{{\mathcal{O}}}}_{k,\omega}$ by skewness as in Remark 9.14. Also as in that remark, skewness also provides a parameterization of the quotient of the pure parameters.

10.5. Sturmian minimal sets, the case $k=1$

We will need the special and much studied case of symbolic kfsm sets for $k=1$ . When $k=1$ there is only one allowable choice for $\nu$ , namely $\nu=1-\omega$ and so we write $C_{1}(\omega)$ for $C_{1}(\omega,1-\omega)=B_{1}(\omega,1-\omega)$ . When $\omega$ is rational $C_{1}(\omega)$ is a single periodic orbit and when $\omega$ is irrational it is a semi-Denjoy minimal set. These minimal sets (and associated sequences) have significant historical importance and an abundance of literature (see [2] for a survey). Their main importance here is as an indicator of when a given number is in the rotation set.

Definition 10.14.

The minimal set $C_{1}(\omega)\subset\Sigma_{2}^{+}$ is called the Sturmian minimal set with rotation number $\omega$ .

To avoid confusion with the many definitions in the literature we note that here “Sturmian” refers to a minimal set and not a sequence and it is subset of the one-sided shift $\Sigma_{2}^{+}$ . The next result is standard and we remark on one proof in Remark 13.6.

Lemma 10.15.

$\omega\in\rho(\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle)$ if and only if $C_{1}(\omega)\subset\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ . If $0\leq\omega_{1}<\omega_{2}\leq 1$ , then in $\Sigma_{2}^{+}$ ,

\min C_{1}(\omega_{1})<\min C_{1}(\omega_{2})<\max C_{1}(\omega_{1})<\max C_{2}(\omega_{2}).

Definition 10.16.

For a fixed $k$ , let $\vec{\nu}_{s}(\omega)$ be defined by $(\vec{\nu}_{s}(\omega))_{i}=1-\omega$ for $i=1,\dots,k$ .

Remark 10.17.

Since $\tau(\vec{\nu}_{s})=\vec{\nu}_{s}$ if follows directly from Lemma 10.7 that for any $k$ , $C_{k}(\omega,\vec{\nu}_{s})=C_{1}(\omega)$ , the Sturmian minimal set with rotation number $\omega$ .

11. Structure of $\operatorname{HM}_{k}(g)$

One obvious property of $\operatorname{HM}_{k}(g)$ is the symmetry $\tau(\operatorname{HM}_{k}(g))=\operatorname{HM}_{k}(g)$ for all $k$ . The full structure of $\operatorname{HM}_{k}(g)$ for a general $g\in{\mathcal{G}}$ is quite complicated and will be saved for future papers. Here we focus on the structure near the diagonal in ${\mathcal{D}}_{k}$ .

11.1. Irrationals on the diagonal

We parameterize the diagonal $\Delta_{k}\subset{\mathcal{D}}_{k}$ by $\omega$ using $\vec{\nu}_{s}(\omega)\in\Delta_{k}$ as defined in the previous section and so

\Delta_{k}=\{\vec{\nu}_{s}(\omega)\colon 0\leq\omega\leq 1\}.

For $g\in{\mathcal{G}}$ the next result asserts that for each irrational $\alpha\in\operatorname{Int}(\rho(g))$ there is some $\delta=\delta(\alpha)$ so that the neighborhood $N_{\delta}(\vec{\nu}_{s}(\alpha))\subset\operatorname{HM}_{k}(g)$ . It gives the proof of Theorem 1.2(a).

Theorem 11.1.

Assume $g\in{\mathcal{G}}$ and $k>0$

(a)

$\operatorname{HM}_{k}(g)\cap\Delta_{k}=\{\vec{\nu}_{s}(\omega)\colon\omega\in\rho(g)\}$
(b)

If $\alpha\not\in{\mathbb{Q}}$ with $\alpha\in\operatorname{Int}(\rho(g))$ , there exists a $\delta>0$ so that $N_{\delta}(\vec{\nu}_{s}(\alpha))\subset\operatorname{HM}_{k}(g)$ .
(c)

If $\alpha\in\operatorname{Int}(\rho(g))\setminus{\mathbb{Q}}$ , then ${\mathcal{O}}_{k}(g)$ contains a $(k-1)$ -dimensional topological disc consisting of unique invariant measures each supported on a member of a family of $k$ -fold semi-monotone semi-Denjoy minimal sets with rotation number $\alpha$ .

Proof.

Assume ${\hat{\Lambda}}_{1}(g)=\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ . For (a) $C_{k}(\omega,\vec{\nu}_{s}(\omega))=C_{1}(\omega)$ , the Sturmian minimal set with rotation number $\omega$ and from Lemma 10.15, $C_{1}(\omega)\subset\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ if and only if $\omega\in\rho(\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle)=\rho(g)$ .

For (b) note that the pair $(\alpha,\vec{\nu}_{s})$ is nonresonant. We will first show that if $\alpha\in\operatorname{Int}(\rho(g))$ then there exists an $\epsilon>0$ , so that $\operatorname{HD}(C_{k}(\alpha,\vec{\nu}_{s}(\alpha)),C_{k}(\omega,\vec{\nu}))<\epsilon$ implies $C_{k}(\omega,\vec{\nu})\subset\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ . Pick $\alpha_{1},\alpha_{2}\in\operatorname{Int}(\rho(g))$ with $\alpha_{1}<\alpha<\alpha_{2}$ . Thus by Lemma 10.15, in $\Sigma_{2}^{+}$

{\underline{\kappa}}_{0}<\min C_{1}(\alpha_{1})<\min C_{1}(\alpha)<\max C_{1}(\alpha)<\max C_{1}(\alpha_{2})<{\underline{\kappa}}_{1}

and let

\epsilon=\min\{d(\min C_{1}(\alpha_{1}),\min C_{1}(\alpha)),d(\max C_{1}(\alpha),\max C_{1}(\alpha_{2}))\}.

Thus $\operatorname{HD}(C_{k}(\alpha,\vec{\nu}_{s}(\alpha)),C_{k}(\omega,\vec{\nu}))=\operatorname{HD}(C_{1}(\alpha),C_{k}(\omega,\vec{\nu}))<\epsilon$ implies that the compact, invariant set $C_{k}(\omega,\vec{\nu})$ satisfies $\min C_{1}(\alpha_{1})<C_{k}(\omega,\vec{\nu})<\max C_{1}(\alpha_{2})$ and so $C_{k}(\omega,\vec{\nu})\subset\langle{\underline{\kappa}}_{0},{\underline{\kappa}}_{1}\rangle$ .

Using the continuity of $C_{k}$ at nonresonant irrationals from Theorem 10.10(a) there is a $\delta>0$ so that $\|(\omega,\vec{\nu})-(\alpha,\vec{\nu}_{s}\|<\delta$ implies $C_{k}(\omega,\vec{\nu})\subset N_{\epsilon}(C_{k}(\alpha,\vec{\nu}_{s})$ , and so $(\omega,\vec{\nu})\in\operatorname{HM}(g)$ .

Since $\tau(N_{\delta}(\vec{\nu}_{s}(\alpha)))=(N_{\delta}(\vec{\nu}_{s}(\alpha)))$ , the neighborhood descends to one in $\overline{\operatorname{HM}}_{k}(g)$ and $\beta_{k}$ is a homeomorphism on irrational slices of $\overline{\operatorname{HM}}_{k}(g)$ (Theorem 10.10(b)) yielding (c). ∎

12. Rational Slices

In this section we study rational slices in the HM parameter and in ${\mathcal{B}}_{k}(g)$ and ${\mathcal{C}}_{k}(g)$ . As proved in Theorem 8.5, each $B_{k}(p/q,\vec{\nu})$ consists of a collection of periodic orbits all of period $qk/\gcd(p,k)$ with the same rotation number and as a set they are kfsm. Note that this is stronger than each periodic orbit being individually kfsm. The invariant measure $\lambda_{k}(p/q,\vec{\nu})$ is a convex combination of the unique measures supported on each periodic orbit.

12.1. periods in $\Omega_{1}$

The next lemma examines how the periods of $B_{k}$ can change after projection to $C_{k}$ via ${\hat{\pi}}_{k}$ .

Lemma 12.1.

Fix $k>0$ and $p/q\in{\mathbb{Q}}$ and assume $\vec{\nu}$ is allowable for $p/q$ . If $\tau^{j}(\vec{\nu})=\vec{\nu}$ with $0<j\leq k$ and it is the least such $j$ , then the period of $C_{k}(p/q,\vec{\nu})$ is $jq/\gcd(j,p)$ .

Proof.

Recall from Theorem 8.5 that the period of $B_{k}(p/q,\vec{\nu})$ is $kq/\gcd(k,p)$ . If $j=k$ by Lemma 10.7(c) ${\hat{\pi}}_{k}:B_{k}(p/q,\vec{\nu})\rightarrow C_{k}(p/q,\vec{\nu})$ is injective and since $\sigma_{1}{\hat{p}}_{k}={\hat{\pi}}_{k}\sigma_{k}$ , $B_{k}(p/q,\vec{\nu})$ and $C_{k}(p/q,\vec{\nu})$ have the same period. Now if $j<k$ by Lemma 10.7(b), $C_{k}(p/q,\vec{\nu})=C_{j}(p/q,\vec{\nu}^{\;\prime})$ where $\vec{\nu}^{\;\prime}=(\nu_{1},\dots,\nu_{j})$ . Since $j$ is the least such, ${\hat{\pi}}_{j}:B_{j}(p/q,\vec{\nu}^{\;\prime})\rightarrow C_{j}(p/q,\vec{\nu}^{\;\prime})$ is injective and $C_{j}(p/q,\vec{\nu}^{\;\prime})$ has period $jq/\gcd(j,p)$ . ∎

12.2. the rational structure theorem

The theorem in this section describes in more detail how the measures on $p/q$ -kfsm vary with the parameter.

In the HM construction fix $k$ , $0<p/q<1$ with $\gcd(p,q)=1$ and an allowable $\vec{\nu}$ . We often suppress dependence on these choices and so $R=R_{p/q}$ , etc. Let $N=qk/\gcd(p,k)$ so $N$ is the period of $R$ acting on $S_{k}$ . Recall the address intervals are $X_{j}=[\ell_{j},r_{j}]$ for $j=0,\dots,2k-1$ and so $r_{j}=\ell_{j+1}$ . The good set is $G$ and the itinerary map is $\zeta$ . When we write $\zeta(x)$ it is implicitly assumed that $x\in G$ .

The orbit of $0$ , $o(0,R)$ , partitions $S_{k}$ into $N$ pieces, each of width $k/N=\gcd(p,k)/q$ . Thus $J=[k-\gcd(p,k)/q,1)$ is a fundamental domain for the action of $R$ on $S_{k}$ in the sense that $S_{k}=\cup_{i=0}^{N-1}R^{i}(J)$ as a disjoint union. Thus for each $0\leq p\leq 2k-1$ there is a unique $0\leq m<N$ with $\ell_{p}\in R^{m}(J)$ , and then let $d_{p}=R^{-m}(\ell_{p})$ . Note that since $|X_{2j+1}|=p/q$ , $d_{2j}=d_{2j-1}$ and that all $d_{2j+1}$ as well as both endpoints of $J$ are not in $G$ . Finally, for $j=0,1,\dots,2k-1$ and $x\in J\cap G$ , let $M_{j}(x)=\{0\leq i<N\colon\zeta(x)_{i}=j\}=\{i\colon R^{i}(x)\in X_{j}\}$ .

Lemma 12.2.

Assume $x,x^{\prime}\in J\cap G$ .
(a)

$\zeta(x)=\zeta(x^{\prime})$ if and only if $M_{2j+1}(x)=M_{2j+1}(x^{\prime})$ for all $j=0,\dots,k-1$ .
(b)

For each $j$ , $M_{2j+1}(x)=M_{2j+1}(x^{\prime})$ if and only if $x$ and $x^{\prime}$ are in the same component of $J-\{d_{2j+1}\}$ .
(c)

For each $k$ , $\#M_{2j}(x)=\#M_{2j}(x^{\prime})$ if and only if $x$ and $x^{\prime}$ are in the same component of $\Sigma\setminus\{d_{2j-1},d_{2j+1}\}$ where $\Sigma$ is the circle $\Sigma=J/\mkern-8.0mu\sim$ with $(k-1/N)\sim k$ .

Proof.

First note that both endpoints of $J$ are not in $G$ so they are out of consideration for $x$ and $x^{\prime}$ in what follows.

For (a) one implication is obvious. For the other, it suffices to we show that the collection of $M_{2j+1}(x)$ determines ${\underline{s}}=\zeta(x)$ . By Remark 7.11 we know that ${\underline{s}}\in\Omega_{k}$ and so its one step transitions are governed by (5.2). If $s_{i}=2j+1$ then $s_{i+1}=2j+2$ or $2j+3$ and we know which depending on whether $i+1\in M_{2j+3}$ or not. Similarly, if $s_{i}=2j$ then $s_{i+1}$ is determined by whether $i+1\in M_{2j+1}$ or not. Thus ${\underline{s}}$ is determined completing the proof of (a).

For (b), first note that $|X_{2k-1}|=p/q$ and $(p/\gcd(p,k))(k/N)=p/q$ . Thus $X_{2k-1}$ is exactly filled with $p/\gcd(p,k)$ iterates of $J$ with disjoint interiors. Thus $M_{2k-1}(x)=M_{2k-1}(x^{\prime})$ for all $x\in J\cap G$ . Thus we only consider $0\leq j<k-1$ . If $i$ is such that $R^{i}(J)\subset X_{2j+1}$ , then $i\in M_{2j+1}(x)$ for all $x\in J$ and if $R^{i}(\operatorname{Int}(J))\cap X_{2j+1}=\emptyset$ then $i\not\in M_{2j+1}(x)$ for all $x\in J\cap G$ . If $\ell_{2j+1}\in R^{i}(\operatorname{Int}(J))$ , then for $x>d_{2j+1}$ in $J$ , then $i\in M_{2j+1}(x)$ and for $x<d_{2j+1}$ , $i\not\in M_{2j+1}(x)$ . The last situation to consider is $r_{2j+1}\in R^{i}(\operatorname{Int}(J))$ . Since $|X_{2k+1}|=p/q$ , we have $R^{i}(d_{2j+1})=r_{2j+1}$ and so then for $x>d_{2j+1}$ in $J$ , then $i\not\in M_{2j+1}(x)$ and for $x<d_{2j+1}$ , $i\in M_{2j+1}(x)$ , completing the proof of (b).

If $d_{2j-1}=d_{2j+1}$ every $x\in\Sigma\setminus\{d_{2j-1}\}$ as the same number of indices in $M_{2j}$ , so assume that $d_{2j-1}<d_{2j+1}$ with the other inequality being similar. If $i$ is such that $R^{i}(J)\subset X_{2j}$ then $i\in M_{2j}(x)$ for all $x\in J$ . If $i$ is such that $R^{i}(J)\cap X_{2j}=\emptyset$ then $i\not\in M_{2j}(x)$ for all $x\in J$ . If $i$ is such that $\ell_{2j}\in R^{i}(J)$ then $i\in M_{2j}(x)$ if and only if $x>d_{2j-1}$ in $J$ . If $i$ is such that $\ell_{2j+1}=r_{2j}\in R^{i}(J)$ then $i\in M_{2j}(x)$ if and only if $x<d_{2j+1}$ in $J$ , finishing the proof. ∎

Corollary 12.3.

If the nonempty connected components of $\Sigma\setminus\cup_{j=0}^{k-1}\{d_{2j+1}\}$ are $K_{1},\dots,K_{m}$ , then $B_{k}(p/q,\nu)$ consists of exactly $m$ distinct periodic orbits $P_{1},\dots,P_{m}$ with $\zeta(x)\in P_{j}$ if and only if $x\in o(K_{j},R)$ . Further, $\lambda_{k}(p/q,\nu)=\sum N|K_{j}|\delta_{j}$ with $\delta_{j}$ the unique invariant probability measure supported in $P_{j}$ and $N=qk/\gcd(p,k)$ .

Proof.

As noted above, $J$ is a fundamental domain for the action of $R$ on $S_{k}$ and so it it suffices to study $\zeta(x)$ for $x\in J$ .

Combining Lemma 12.2(a) and (b) we have that for $x\in\Sigma$ , $\zeta(x)=\zeta(x^{\prime})$ if and only if $x$ and $x^{\prime}$ are in the same component $K_{j}$ . Further, using Lemma 12.2(c), $\zeta(x)$ and $\zeta(x^{\prime})$ can be on the same $\sigma$ -orbit if and only if they are in the same component $K_{j}$ , proving the first sentence of the theorem. The second sentence follows from the definition of $\lambda_{k}$ , the fact that $S_{k}=\cup_{i=1}^{N}R^{i}(J)$ , and $R$ preserves Lebesgue measure. ∎

12.3. The pure lattice and the structure of $\operatorname{HM}_{kp/q}$

We now describe the pure affine lattice in more detail with an eye towards counting the number of $p/q$ -periodic kfsm sets. For this a new method of specifying the address system in $S_{k}$ will be useful. We fix a $k$ and an $\omega=p/q$ and sometimes suppress dependence on them

Recall that a pair $(p/q,\vec{\nu})$ specifies an address system $\{X_{j}(p/q,\vec{\nu})\}$ with each $X_{j}(p/q,\vec{\nu})=[\ell_{j},r_{j}]$ . For each $i=1,\dots,k-1$ let $\delta_{i}$ be the signed displacement of the address system from its totally symmetric position given by $(\omega,\vec{\nu}_{s}(\omega))$ . Thus

\delta_{i}(\vec{\nu})=(\nu_{1}+\dots\nu_{i})-i(1-\omega).

(12.1)

Since in the HM-construction $X_{2k}$ is fixed for all $\vec{\nu}$ , the vector $\vec{\delta}(\omega)$ is $(k-1)$ -dimensional and so $\vec{\delta}:{\mathcal{D}}_{k,p/q}\rightarrow\vec{\delta}({\mathcal{D}}_{k,p/q})$ is an affine map from the simplex $\sum\nu_{i}=k(1-p/q)$ to a subset of ${\mathbb{R}}^{k-1}$ . Note that $\vec{\delta}(\vec{\nu}_{s})=\vec{0}$ .

Lemma 12.4.

Given $k$ and $p/q$ there exists a $\vec{\eta}$ with $\|\vec{\eta}\|_{\infty}\leq\gcd(p,k)/(2q)$ so that $\vec{\nu}\in{\mathcal{D}}_{k,p/q}$ is a pure parameter for $p/q$ if and only if $\vec{\delta}(p/q,\vec{\nu})\in\vec{\eta}+(\gcd(p,k)/q){\mathbb{Z}}^{k-1}$ in $\vec{\delta}({\mathcal{D}}_{k,p/q})$ .

Proof.

The structure theorem Theorem 12.3 implies that $B_{k}(p/q,\vec{\nu})$ is a single periodic orbit if and only if no $d_{2j-1}$ is in the interior of $J$ . This happens if and only if all $\ell_{2j-1}$ are contained in $o(0,R_{p/q})$ . Now $o(0,R_{p/q})$ divides $S_{k}$ evenly into subintervals of length $\gcd(p,k)/q$ . For each $j=1,\dots,k-1$ let $m_{j}$ be such that $R^{m_{j}}_{p/q}(0)$ is the point on $o(0,R_{p/q})$ that is closest to $\ell_{2j-1}$ and define $\eta_{j}=\ell_{2j-1}-R^{m_{j}}_{p/q}(0)$ . Thus $\|\vec{\eta}\|_{\infty}\leq\gcd(p,k)/(2q)$ and $\vec{\nu}$ is pure if and only if $\phi(\vec{\nu})\in\vec{\eta}+(\gcd(p,k)/q){\mathbb{Z}}^{k-1}$ . ∎

Definition 12.5.

The set $L=\vec{\eta}+(\gcd(p,k)/q){\mathbb{Z}}^{k-1}\cap\vec{\delta}({\mathcal{D}}_{k,p/q})$ is called the $p/q$ -pure affine lattice as is its pre-image $\vec{\delta}^{-1}(L)\subset{\mathbb{R}}^{n-1}$

12.4. Sub-resonance and the size of clusters

Definition 12.6.

When $\omega=p/q$ , the pair $(p/q,\nu)$ is called sub-resonant if for some $qk/\gcd(p,k)>n>1$ and $j\not=j^{\prime}$ , $R_{\omega}^{n}(\ell_{j})=\ell_{j^{\prime}}$ .

It follows from Theorem 12.3 that the number of sub-resonances in a $(p/q,\vec{\nu})\in{\mathcal{D}}_{k,p/q}$ controls the number of distinct periodic orbits in a cluster $B_{k}(p/q,\vec{\nu})$ with no sub-resonance corresponding to $k$ distinct periodic orbits and all the $\ell_{j}$ on a single $R_{p/q}$ orbit corresponding to $B_{k}(p/q,\vec{\nu})$ being a single periodic orbit so $(p/q,\vec{\nu})$ is a pure parameter

The set sub-resonance parameters is a finite collection of codimension one affine subspaces in ${\mathcal{D}}_{k,p/q}$ and thus the no sub-resonance case is open, dense and full measure in ${\mathcal{D}}_{k,p/q}$ . Thus in $\operatorname{HM}_{k,p/q}$ the typical parameter corresponds to a cluster of $k$ periodic orbits. It also follows that the assignment $\vec{\nu}\mapsto B_{k}(p/q,\vec{\nu})$ restricted to $\operatorname{HM}_{k,p/q}$ is constant and thus continuous on connected components of the no sub-resonance parameters and is discontinuous at the sub-resonance parameters.

12.5. Estimating the number of $p/q$ kfsm sets

For a given $g\in{\mathcal{G}}$ the number of points from the pure $p/q$ -lattice $\operatorname{Pure}_{k,p/q}$ contained in $\operatorname{HM}_{k,p/q}(g)$ tells us how many distinct periodic orbits there are in ${\hat{{\mathcal{B}}}}_{k}(g)$ . So by Lemma 10.7 it yields how many distinct periodic $p/q$ -kfsm sets $g$ has. We get an estimate for this number using the continuity properties of $B_{k}$ from Theorem 9.5 and the relationship of kfsm sets in $S_{k}$ to those in $S^{1}$ . The next result proves Theorem 1.2(b).

Theorem 12.7.

If $\alpha\in\operatorname{Int}(\rho(g))$ , $\alpha\not\in{\mathbb{Q}}$ , $k>0$ and $p_{n}/q_{n}$ is a sequence of rationals in lowest terms with $p_{n}/q_{n}\rightarrow\alpha$ , then there exists a $C>0$ so that for sufficiently large $n$ the number of distinct periodic $p_{n}/q_{n}$ kfsm sets in $\Lambda_{1}(g)$ is greater than or equal to $Cq_{n}^{k-1}$ .

Proof.

By Theorem 9.5(b) there is an $\epsilon_{1}>0$ so that $N_{\epsilon_{1}}(\vec{\nu}_{s}(\alpha))\subset\operatorname{HM}_{k}(g)$ where recall that $\vec{\nu}_{s}(\alpha)$ is the Sturmian $\vec{\nu}$ for $\alpha$ on the diagonal of ${\mathcal{D}}_{k}$ . Since $\vec{\delta}$ is a homeomorphism there is an $\epsilon$ -ball $H$ in the max norm with $\epsilon>0$ about $\vec{0}$ in $\vec{\delta}({\mathcal{D}}_{k})$ with $\vec{\delta}^{-1}(H)\subset\operatorname{HM}_{k}(g)$ . Thus if $|p_{n}/q_{n}-\alpha|<\epsilon$ there is a $\epsilon$ -ball in the max norm, i.e., a $(k-1)$ -dimensional hypercube $H_{1}$ , about $(p_{n}/q_{n},\vec{0})$ in $\vec{\delta}({\mathcal{D}}_{k,p_{n}/q_{n}})$ with $\vec{\delta}^{-1}(H_{1})\subset\operatorname{HM}_{k,p_{n}/q_{n}}(g)$ .

We next estimate the number of pure resonance $\vec{\nu}$ in $H_{1}$ . By Lemma 12.4, the pure $\vec{\nu}$ form an affine lattice with linear separation $\gcd(p_{n},k)/q_{n}$ . Thus for $p_{n}/q_{n}$ close enough to $\alpha$ , the number of lattice points in $H_{1}$ is larger than

\left(\frac{\epsilon q_{n}}{\gcd(p_{n},k)}\right)^{k-1}\geq\left(\frac{\epsilon q_{n}}{k}\right)^{k-1}.

since $\gcd(p_{n},k)\leq k$ . Thus since $\vec{\delta}$ is a homeomorphism the same estimate holds for the number of pure lattice points in $\vec{\delta}^{-1}(H_{1})\subset\operatorname{HM}_{k,p_{n}/q_{n}}(g)$ . By Theorem 9.5 this tells us how many distinct periodic $p_{n}/q_{n}$ are in ${\hat{{\mathcal{B}}}}_{k}(g)$ and thus in ${\mathcal{B}}_{k}(g)$ by Theorem 7.6.

To project this estimate to kfsm sets in $S^{1}$ , recall from Theorem 10.10 that $\theta_{k}:\overline{\operatorname{HM}}_{k,p_{n}/q_{n}}(g)\rightarrow{\mathcal{C}}_{k,p_{n}/q_{n}}(g)$ is injective on the pure lattice. The projection $\operatorname{Pure}(k,p_{n}/q_{n})\rightarrow\overline{\operatorname{Pure}}(k,p_{n}/q_{n})$ is at most $k$ to $1$ and so the number of distinct $p_{n}/q_{n}$ periodic orbits in ${\mathcal{C}}(g)$ is greater than or equal to

\frac{1}{k}\left(\frac{\epsilon}{k}\right)^{k-1}q_{n}^{k-1}.

∎

Remark 12.8.

For a pure $\vec{\nu}$ for $p/q$ when $\tau^{j}(\vec{\nu})=\vec{\nu}$ for some $0<j<k$ the period of the $C_{k}(p/q,\vec{\nu})$ counted in the theorem is $jq/\gcd(j,p)$ (Lemma 12.1). In the typical case of no such symmetry the period is $kq/\gcd(k,p)$ . So, for example, when $p$ and $k$ are relatively prime, the counted periodic orbit has rotation type $(pk,qk)$ and when $k$ divides $p$ the rotation type is $(p,q)$ . By making judicious choices of the sequence $p_{n}/q_{n}\rightarrow\alpha$ , one can control the rotation types of the counted periodic orbits.

13. Parameterization via the interpolated family of maps

We return now to the heuristic description in the introduction of kfsm sets via a family of interpolated semi-monotone maps and prove a few results and connections to the HM-parameterization. Since we are mainly developing a heuristic, some details are left to the reader. In many ways this point of view is better for studying kfsm sets while the HM construction is better for measures. Initially the parameterization depends on the map $\tilde{g}\in{\mathcal{G}}$ but using the model map we will get a uniform parameterization.

13.1. The family of k-fold interpolated maps for $g\in{\mathcal{G}}$

Fix $g\in{\mathcal{G}}$ with preferred lift $\tilde{g}$ . For $y\in[g(x_{min}+n),g(x_{max}+n)]$ there is a unique $x\in[x_{min}+n,x_{max}+n]$ with $\tilde{g}(x)=y$ . Denote this $x$ as $b_{n}(y)$ ( $b$ for branch). Let $L_{g}=\tilde{g}(\min(\Lambda_{\infty}(g)\cap I_{0}))$ and $U_{g}=\tilde{g}(\max(\Lambda_{\infty}(g)\cap I_{-1}))$ with the $I_{i}$ as defined in Section 5.1. Note from the definition of the class ${\mathcal{G}}$ , $0\leq L_{g}<U_{g}\leq 1$ and by equivariance, $L_{g}+j=\tilde{g}(\min(\Lambda_{\infty}(g)\cap I_{2j}))$ and $U_{g}+j=\tilde{g}(\max(\Lambda_{\infty}(g)\cap I_{2j-1}))$ .

Definition 13.1.

For $\vec{c}\in{\mathbb{R}}^{k}$ define $\tilde{c}\in{\mathbb{R}}^{k}$ via $\tilde{c}_{j}=c_{j}+j-1$ for $j=1,\dots,k$ .

Fix $k>0$ . For $\vec{c}\in[L_{g},U_{g}]^{k}$ and for $j=1,2,\dots,k$ define $\tilde{H}_{k\vec{c}}(x)$ on $[b_{-1}(0),b_{-1}(0)+k]$ as

\tilde{H}_{k\vec{c}}(x)=\begin{cases}\tilde{c}_{j}&\text{when}\ x\in[b_{j-2}(c_{j}),b_{j-1}(c_{j})]\\ \tilde{g}(x)&\text{otherwise}\end{cases}

and extend to $\tilde{H}_{k\vec{c}}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ so that $\tilde{H}_{k\vec{c}}(x+k)=\tilde{H}_{k\vec{c}}(x)+k$ . See Figure 1. Next define $H_{k\vec{c}}:S_{k}\rightarrow S_{k}$ as the descent of $H_{k,\vec{c}}$ to $S_{k}$ .

Example: The model map For the model map $f_{m}$ , $x_{min}=0,x_{max}=1/2,L=0,U=1/2$ and $b_{j}(y)=(y+2j)/3$ .

Given a compact $Z\subset\Lambda_{k}(g)$ , for $j=0,\dots,k$ let

\ell_{j}^{\prime}(Z)=\tilde{g}_{k}(\max\{Z\cap I_{2j-1}\})-(j-1)\ \text{and}\ r_{j}^{\prime}(Z)=\tilde{g}_{k}(\min\{Z\cap I_{2j}\})-(j-1).

If for some $j$ we have $\ell_{j}^{\prime}<L_{g}$ let $\ell_{j}=L_{g}$ otherwise let $\ell_{j}=\ell_{j}^{\prime}$ . Similarly, and if for some $j$ , we have $r_{j}^{\prime}>R_{g}$ let $r_{j}=R_{g}$ otherwise let $r_{j}=r_{j}^{\prime}$ . Not that these $r$ ’s and $\ell$ ’s are unrelated to those in Section 9.1.

Theorem 13.2.

Assume $Z\subset\Lambda_{k}(g)$ is compact and invariant. The following are equivalent

(a)

$Z$ is a kfsm set
(b)

For $j=1,\dots,k$ , $\ell_{j}(Z)\leq r_{j}(Z)$ .
(c)

$Z\subset P(H_{k\vec{c}})$ for

$\vec{c}\in\prod_{j=1}^{k}[\ell_{j}(Z),r_{j}(Z)]$ (13.1)

thus $(\tilde{g}_{k})_{|Z}=(H_{k\vec{c}})_{|Z}$ .

Proof.

If for some $j$ , $\ell_{j}(Z)>r_{j}(Z)$ then $g$ restricted to $Z$ doesn’t preserve the cyclic order, and so (a) implies (b). (c) implies (a) since invariant sets in nondecreasing maps are always kfsm. Finally, (b) says that $Z\subset P(H_{k\vec{c}})$ for $\vec{c}$ in the given range. ∎

Definition 13.3.

For $Z\in{\mathcal{B}}_{k}(g)$ , let

\operatorname{Box}_{g}(Z)=\prod_{j=1}^{k}[\ell_{j}(Z),r_{j}(Z)]

and so $\operatorname{Box}_{g}(Z)\subset[L_{g},U_{g}]^{k}$ .

Remark 13.4.

(a)

Nothing in the theorem requires $Z$ to be recurrent. If it is, so $Z\in{\mathcal{B}}_{k}(g)$ , from Theorem 8.5, $\iota_{k}(Z)=B_{k}(\omega,\vec{\nu})$ where $\omega=\rho_{k}(Z)$ and $\vec{\nu}\in{\mathcal{D}}_{k\omega}$ .
(b)

When $Z$ is a periodic orbit or cluster, $\operatorname{Box}_{g}(Z)$ is $k$ -dimensional. When $Z$ is a periodic orbit cluster its box is equal to the intersections of the boxes of its constituent single periodic orbits.
(c)

When $Z$ is a semi-Denjoy minimal set contained in $P(H_{k\vec{c}})$ recall that a tight flat spot of $Z$ is one for which both endpoints of a flat spot of $H_{k\vec{c}}$ are in $Z$ . The dimension of $\operatorname{Box}_{g}(Z)$ is the same as the number of loose flat spots in $Z$ . Since by Lemma 3.5(b), $Z$ cannot have $k$ tight flat spots, the dimension of $\operatorname{Box}_{g}(Z)$ is between $0$ and $k-1$ .
(d)

Note that in contrast, in the HM parameterization, each single periodic orbit or semi-Denjoy minimal corresponds to just one point.
(e)

When $\alpha\not\in{\mathbb{Q}}$ , if $Z=\iota_{k}^{-1}(B_{k}(\alpha,\vec{\nu}))$ from the HM construction then the number of loose flat spots of $Z$ is the same as the number of resonances of $(\alpha,\vec{\nu})$ i.e., $j\not=j^{\prime}$ with $R_{\alpha}^{N}(\ell_{j})=\ell_{j^{\prime}}$ for some $n>1$ which is then the same as the dimension of $\operatorname{Box}_{g}(Z)$ .

13.2. Nonrecurrence and kfsm sets that hit the negative slope region

Throughout this paper we have assumed that the kfsm sets were recurrent and avoided the negative slope region. In this section we use the interpolated maps to motivate these assumptions.

Assume now that $Z$ is a kfsm set for some $g\in{\mathcal{G}}$ and $Z$ contains points in the negative slope region of $g$ . It still follows that $Z$ is an invariant set of some $H_{k\vec{c}}$ . Let $Z^{\prime}$ be the maximal recurrent set in $P(H_{k\vec{c}})$ . A gap of $Z^{\prime}$ is a component of the complement of $Z^{\prime}$ that contains a flat spot of $H_{k\vec{c}}$ . In formulas, a gap is an interval $(\max\{Z^{\prime}\cap I_{2j-1}\},\min\{Z^{\prime}\cap I_{2j}\})$ for some $j$ . Since $g_{k}$ acting on $Z$ is semi-monotone, $Z$ can contain at most one point $p_{j}$ in the negative slope region within each gap.

There are two cases. In the first, which may happen for both rational and irrational rotation number, for all $j^{\prime}$ there is some $n$ so that $f^{i}(p_{j^{\prime}})\not\in\{p_{j}\}$ for all $i\geq n$ . This implies that $H_{k\vec{c}}^{i}(p_{j})\in P(H_{k\vec{c}})$ for all $i\geq n$ and so by Lemma 3.5(c), there is an $n^{\prime}$ so that $H_{k\vec{c}}^{i}(p_{j})\in Z^{\prime}$ for all $i\geq n^{\prime}$ . Thus in this case negative slope orbits add no additional recurrent dynamics. In Figure 4 the disks give part of a periodic kfsm set and the squares show additional homoclinic points to this kfsm set in the negative and positive slope region.

The second case holds for just rational rotation number and is when some $p_{j}$ is a periodic point; this does add new recurrent kfsm sets. Since the periodic points on the endpoint of a gap always return, there is, in fact, a periodic point in the negative slope region of each gap of $Z^{\prime}$ . By adjusting $\vec{c}$ we can assume that all these gap periodic orbits are also superstable periodic orbits of $H_{k\vec{c}}$ . Since the periodic points in $Z^{\prime}$ are all unstable the periodic points of $Z^{\prime}$ must alternate with these gap periodic points. In particular, the number of gap periodic orbits equals the number of periodic orbits in $Z^{\prime}$ . Thus the addition of the negative slope periodic orbits just adds a factor of two to the basic estimates of Section 12.5.

We again use Figure 4 but this time to discuss the holes in the space of recurrent kfsm sets. Let $H_{c_{0}}$ be an interploted map whose flat spot contains the homoclinic points indicated by squares. Assume we are in the $k=1$ case and so each interpolated map $H_{c}$ contains exactly one recurrent semi-monotone set $Z_{c}$ in its positive slope region. As $c_{n}$ increases to $c_{0}$ the sets $Z_{c_{n}}$ converge in the Hausdorff topology to not just $Z_{c_{0}}$ but to that set union the boxed point in the positive slope region. A similar phenomenon happens as $c_{n}$ decreases to $c_{0}$ . This phenomenon also clearly happens for loose gaps of semi-Denjoy minimal sets and happens for all $k$ . This is the geometric explanation of the holes in ${\mathcal{B}}_{k}(g)$ and the discontinuity of $B_{k}$ discussed in Section 9.1.

13.3. The rotation number diagram

Fix $g\in{\mathcal{G}}$ . Since $H_{k\vec{c}}:S_{k}\rightarrow S_{k}$ we define $R_{k}(\vec{c})=k\rho(D_{k}\circ H_{k\vec{c}}\circ D_{k}^{-1})$ . Thus if $Z\subset P(H_{k\vec{c}})$ is compact invariant then $\rho_{k}(Z)=R_{k}(\vec{c})$ . We treat $R_{k}$ as a function $R_{k}:[L_{g},U_{g}]^{k}\rightarrow{\mathbb{R}}$ .

Let ${\mathbb{R}}_{+}^{k}=\{\vec{u}\in{\mathbb{R}}_{K}\colon\ \text{all}\ u_{i}>0\}$ . The open projective positive cone in ${\mathbb{R}}^{k}$ is $Q_{k}=\{\vec{u}\in{\mathbb{R}}^{k}_{+}\colon\|\vec{u}\|_{2}=1\}$ . For a given $k,\omega$ define $\varphi_{-,\omega},\varphi_{+,\omega}:Q_{k}\rightarrow{\mathbb{R}}^{+}$ as

	$\displaystyle\varphi_{-,\omega}(\vec{u})$	$\displaystyle=\min\{t\in{\mathbb{R}}^{+}\colon R_{k}(t\vec{u}+\vec{L})=\omega\}$
		$\displaystyle\varphi_{+,\omega}(\vec{u})=\max\{t\in{\mathbb{R}}^{+}\colon R_{k}(t\vec{u}+\vec{L})=\omega\}$

where $\vec{L}=(L_{g},L_{g},\dots,L_{g})$ . So $\varphi_{-,\omega}$ and $\varphi_{+,\omega}$ give the top and bottom edges of the level set $R_{k}^{-1}(\omega)$ when view from the origin.

Theorem 13.5.

Assume $g\in{\mathcal{G}}$ and construct $R_{k}$ as above

(a)

$R_{k}$ is continuous function and is nondecreasing in $t$ along any line $\vec{c}=t\vec{u}+\vec{v}$ with all $v_{i}\geq 0$ .
(b)

For all $\omega$ the functions $\varphi_{-,\omega}$ and $\varphi_{+,\omega}$ are continuous.
(c)

For rational $\omega$ , $\varphi_{-,p/q}<\varphi_{+,p/q}$ while for $\alpha\not\in{\mathbb{Q}}$ then $\varphi_{-,\alpha}=\varphi_{+,\alpha}$ . Thus each level set $R_{k}^{-1}(p/q)$ is homeomorphic to a $(k-1)$ -dimensional open disk product a nontrivial closed interval while each $R_{k}^{-1}(\alpha)$ is homeomorphic to a $(k-1)$ -dimensional open disk.
(d)

$\rho(g)=[\rho(H_{L_{g}}),\rho(H_{U_{g}})]=\rho(\Lambda_{1}(g),g)$

Proof.

Part (a) follows directly from Lemma 3.1(b) and (c). For (b) assume to the contrary that $\varphi_{-}$ is not continuous. Then there is a sequence $\vec{u}_{n}\rightarrow\vec{u}_{0}$ with $\varphi_{-}(\vec{u}_{n})\not\rightarrow\varphi_{-}(\vec{u}_{0})$ . Passing to a subsequence if necessary, there is some $t_{0}$ with $\varphi_{-}(\vec{u}_{n})\vec{u}_{n}+\vec{L}\rightarrow t_{0}\vec{u}_{0}+\vec{L}$ . By the continuity of $R_{k}$ , $R_{k}(t_{0}\vec{u}_{0}+\vec{L})=\omega$ and by the nonconvergence assumption, there is some $t^{\prime}<t_{0}$ with $R_{k}(t^{\prime}\vec{u}_{0}+\vec{L})=\omega$ . Thus again by the continuity of $R_{k}$ for $n$ large enough there is some $t^{\prime\prime}_{n}<\varphi_{-}(\vec{u}_{n})$ with $R_{k}(t^{\prime\prime}_{n}\vec{u}_{n}+\vec{L})=\omega$ , a contradiction. Therefore, $\varphi_{-}$ is continuous: the continuity of $\varphi_{+}$ is similar.

For (c), pick any $t_{0}$ and $\vec{u}_{0}$ with $R_{k}(t_{0}\vec{u}_{0}+\vec{L})=p/q$ and let $\vec{c}=t_{0}\vec{u}_{0}+\vec{L}$ . Then by Lemma 3.5, $H_{\vec{c}}$ has a periodic orbit $Z\subset P(H_{\vec{c}})$ . Since $Z$ is a finite set there is a nontrivial interval $I$ so that $t\in I$ implies $R_{k}(t\vec{u}_{0}+\vec{L})=p/q$ and so $\varphi_{-,p/q}<\varphi_{+,p/q}$ .

To complete (c), assume to the contrary that for some $\vec{u}_{0}$ , $\varphi_{-,\alpha}(\vec{u}_{0})<\varphi_{+,\alpha}(\vec{u}_{0})$ . Thus by the continuity of $R_{k}$ there is an open ball $N\subset R_{k}^{-1}(\alpha)$ . Pick $\vec{c}\in N$ and let $Z$ be the semi-Denjoy minimal set in $P(H_{\vec{c}})$ guaranteed by Lemma 3.5 which has at least one tight gap, say the gap associated with $c_{1}$ the first coordinate of $\vec{c}$ . Let $y$ be the $x$ -coordinate of the right hand endpoint of this gap and so $\tilde{g}_{k}(y)=c_{1}$ . Since $Z$ is minimal under $H_{\vec{c}}$ there are points $z\in Z$ with $z>y$ and arbitrarily close to $y$ which have a $n>0$ with $y<H_{\vec{c}}^{n}(z)<z$ . Now let $c_{1}^{\prime}=\tilde{g}_{k}(z)$ and $\vec{c}^{\prime}=(c_{1}^{\prime},c_{2},\dots,c_{k})$ and we have that $H^{n}_{\vec{c}^{\prime}}(z)<z$ which says that the $n^{th}$ iterate of the first coordinate flat spot of $H_{\vec{c}^{\prime}}$ is in that flat spot. Thus $H_{\vec{c}^{\prime}}$ has a periodic orbit and so $R_{k}(\vec{c}^{\prime})\not=\alpha$ for some $\vec{c}^{\prime}$ arbitrarily close to $\vec{c}$ , a contradiction.

For (d), assume $k=1$ and $\rho(g)=[\rho_{1},\rho_{2}]$ . Let $H_{T}$ be the semi-monotone map constructed from $g$ to have a single flat spot of height $\tilde{g}(x_{max})$ and $H_{B}$ similarly constructed having a single flat spot of height $\tilde{g}(x_{min})$ . Since $H_{T}\geq\tilde{g}$ , we have $\rho(H_{T})\geq\rho_{1}$ . Now by Lemma 3.5, there is a compact invariant $Z\subset P(H_{T})$ and so $g_{|Z}=(H_{T})_{|Z}$ and so $\rho(H_{T})=\rho(Z,g)\in\rho(g)$ and so $\rho(H_{T})=\rho_{1}$ . Similarly, $\rho(H_{B})=\rho_{2}$ . Note that by definition of $H_{U}$ , the compact invariant $Z\subset P(H_{T})$ also satisfies $Z\subset P(H_{U})$ and so $\rho(H_{T})=\rho(H_{U})$ . Similarly, $\rho(H_{B})=\rho(H_{L})$ . Thus $\rho(g)=[\rho(H_{L}),\rho(H_{U})]$ . Finally, consider the entire family $H_{c}$ for $c\in[L,U]$ . Since $\rho(H_{c})$ is continuous in $c$ , for each $\omega\in[\rho(H_{L}),\rho(H_{U})]$ there is a $c$ with $\rho(H_{c})=\omega$ . Further, for each $c$ there is a compact invariant $Z_{c}\subset P(H_{c})$ and $Z_{c}\subset\Lambda_{1}(g)$ and thus $\omega\in\rho(\Lambda_{1}(g))\subset\rho(g)$

∎

Remark 13.6.

(a)

Note that $H_{T}(x)\leq x+1$ and $H_{B}(x)\geq x$ and thus $g\in{\mathcal{G}}$ implies $\rho(g)\subset[0,1]$ . Further, it follows from (d) that the image of each $R_{k}$ is $\rho(g)$ .
(b)

Part (b) deals only with the part of the level sets of $R_{k}$ in the open set $(L_{g},U_{g})^{k}$ . The extension to all of $[L_{g},U_{g}]^{k}$ is technical and not very illuminating so we leave it to the interested reader.
(c)

Let $\tau$ act on $\vec{c}$ as the left cyclic shift. It easily follows that $R_{k}(\tau(\vec{c}))=R_{k}(\vec{c})$ .
(d)

When $k=1$ there is a one-dimensional family $H_{c}$ for $c\in[L_{g},U_{g}]$ . The rotation number $R_{1}(c)$ is nondecreasing in $c$ and assumes each irrational value at a point and each rational value on an interval by (a) and (c). For each $c$ there is a unique recurrent $Z_{c}\subset P(H_{c})$ and $\iota_{1}(Z_{c})$ is the Sturmian with the given rotation number. This along with the geometry of the family $H_{c}$ give the proof of Lemma 10.15.

13.4. Comparing $g\in{\mathcal{G}}$ to the model map

In this section we use the interpolation parameter $\vec{c}$ to parameterize the $Z\in{\mathcal{B}}_{k}(g)$ for a general $g\in{\mathcal{G}}$ . Notice that for the model map, $\Lambda_{k}(f_{m})$ is all of $\Omega_{k}$ . Thus ${\hat{{\mathcal{B}}}}_{k}(g)\subset{\hat{{\mathcal{B}}}}_{k}(f_{m})$ and so we can pass back to ${\mathcal{B}}(g)$ using the inverse of the itinerary map. Thus we can use a subset of the interpolation parameters of the model map to parameterize ${\mathcal{B}}(g)$ using the symbolic representation of a kfsm set as the link. This subset turns out to be a square of the form $[L^{\prime},U^{\prime}]^{k}$ . In this section we often add an additional subscript of $f$ or $g$ to indicate which map $f_{m}$ or $g$ is involved

Since $\iota_{k,g}(\Lambda_{k}(g))={\hat{\Lambda}}_{k}(g)\subset\Omega_{k}={\hat{\Lambda}}_{k}(f)$ we may define $\psi^{\prime}:\Lambda_{k}(g)\rightarrow\Lambda_{k}(f)$ by $\psi^{\prime}=\iota_{kf}\circ\iota_{kg}^{-1}$ . By Theorem 6.1 $\psi^{\prime}$ is an orientation preserving homeomorphism onto its image as well as a conjugacy. It thus induces a map $\overline{\psi}:{\mathcal{B}}_{k}(g)\rightarrow{\mathcal{B}}_{k}(f)$ .

Recall that the parameters for the model map are $[L_{f},U_{f}]^{k}=[0,1/2]^{k}$ . For a map $\phi:[a,b]\rightarrow[a,b]$ extend it to the Cartesian product as $\phi^{(k)}=(\phi,\phi,\dots,\phi)$ .

Theorem 13.7.

Given $g\in{\mathcal{G}}$ and $k>0$ construct the interpolation parameters $[L_{g},U_{g}]$ . There exists an interval $[L^{\prime},U^{\prime}]\subset[0,1/2]$ and an orientation preserving homeomorphism $\phi:[U_{g},L_{g}]\rightarrow[L^{\prime},U^{\prime}]$ so that for all $Z\in{\mathcal{B}}_{k}(g)$ , $\phi^{(k)}(\operatorname{Box}_{g}(Z)))=\operatorname{Box}_{f}(\overline{\psi}(Z))$ and for all $\omega\in\rho(g)$ , $\phi^{(k)}\rho_{k,g}^{-1}(\omega)=\rho_{k,f}^{-1}(\omega)$ . Where

Proof.

Construct $\psi^{\prime}$ as above. Its properties imply that

\psi^{\prime}(\ell_{j}(Z))=\ell_{j}(\overline{\psi}(Z))\ \text{and}\ \psi^{\prime}(r_{j}(Z))=r_{j}(\overline{\psi}(Z))

(13.2)

for all $Z\subset{\mathcal{B}}_{k}(g)$ and $j=1,\dots,k$ . Let $L^{\prime}=\psi^{\prime}(L_{g})$ and $U^{\prime}=\psi^{\prime}(U_{g})$ . Then $\psi^{\prime}$ restricts to $\psi:\Lambda_{k}(g)\cap[L_{g},U_{g}]^{k}\rightarrow\Lambda_{k}(f)\cap[L^{\prime},U^{\prime}]^{k}$ . Since $\psi T_{k}=T_{k}\psi$ and $\Lambda_{k}(g)\cap[L_{g},U_{g}]^{k}$ is compact we can extend $\psi$ equivariantly to a homeomorphism $\Psi:[L_{g},U_{g}]^{k}\rightarrow[L^{\prime},U^{\prime}]^{k}$ which using (13.2) satisfies $\Psi\circ\operatorname{Box}_{g}=\operatorname{Box}_{f}\circ\overline{\psi}$ . Finally, since $\Psi\circ\tau=\tau\circ\Psi$ (recall $\tau$ is the left cyclic shift) there is a $\phi:[U_{g},L_{g}]\rightarrow[L^{\prime},U^{\prime}]$ with $\Psi=\phi^{(k)}$ . ∎

Thi result implies that the $\rho_{k}$ -diagram for $g$ looks like $k$ -dimensional cube cut from inside the $\rho_{k}$ -diagram of the model map and perhaps rescaled.

13.5. The case $k=2$ : numerics

Figure 5 shows the $k=2$ rotation number diagram for the model map $f_{m}$ . Each connected union of rectangles is the level set of some rational. The rationals with denominator less than $6$ are shown. Only the center rectangle is labeled for each rational. In the figure each rectangle corresponds to a different $2$ -fold semi-monotone periodic orbit. The intersections of these rectangles correspond to $H_{\vec{c}}$ which have a cluster of two periodic orbits

The computation of this diagram used a discrete version of the HM construction. The construction depends on integers $p,q,\mu$ with $0<p/q<1$ , $p$ and $q$ relatively prime, and $0\leq\mu\leq 2(q-p)$ . The discrete circle is the finite cyclic group ${\mathbb{Z}}/2q{\mathbb{Z}}={\mathbb{Z}}_{2q}$ and it is acted on by $R_{p}:n\mapsto n+p$ . The address intervals are $X_{0}^{\prime}=[1,\mu],X_{1}^{\prime}=[\mu+1,\mu+p],X_{2}^{\prime}=[\mu+p+1,2q-p]$ , and $I_{3}=[2q-p+1,2(q-p)]$ . Let $B^{\prime}(p,q,\mu)$ be the itinerary of the point $1$ under $R_{p}$ .

Using Theorem 8.5(b), when $p$ is odd, $R_{p}$ has a single period $2q$ orbit in ${\mathbb{Z}}_{2q}$ . Expanding the points in ${\mathbb{Z}}_{2q}$ to intervals in the circle as in the proof of Theorem 8.5(c), we see that by varying $\mu$ the construction generates all the symbolic $p/q$ -periodic $2$ -fold semi-monotone sets in $\Omega_{2}$ .

Now when $p$ is even, $R_{p}$ has a pair of period $q$ orbits. When $\mu$ is odd, these generate different periodic orbits $B^{\prime}(p,q,\mu)$ . However, $\mu$ even corresponds to a pure parameter and so varying $\mu$ through the even $\mu$ generates all the symbolic $p/q$ -periodic $2$ -fold semi-monotone sets in $\Omega_{2}$ .

The next step is to use $B^{\prime}(p,q,\mu)$ to compute its symbolic box as in Corollary 14.1 below. Finally, we take the inverse of the itinerary map for the model map to get a box in the $\vec{c}$ parameter. Because the map $f_{m}$ has uniform slope of three in its positive slope region the formula for this inverse is ${\underline{s}}\in\Sigma_{2}^{+}$ ,

\iota_{1}^{-1}({\underline{s}})=\sum_{j=0}^{\infty}\frac{s_{j}}{3^{j+1}}.

(13.3)

14. Symbolic kfsm sets and the map $z\mapsto z^{n}$

Using the model map Theorem 13.2 characterising “physical” kfsm sets can be directly transformed into a characterization of symbolic kfsm sets. For compact ${\hat{Z}}\subset\Omega_{k}$ for $j=1,\dots,k$ define

{\hat{\ell}}_{j}(Z)=\sigma_{k}(\max\{{\hat{Z}}\cap[2j-1]\})\ \text{and}\ {\hat{r}}_{j}(Z)=\sigma_{k}(\min\{{\hat{Z}}\cap[2j]\})

Since $\iota_{k}$ is order preserving and onto for the model map we have

Corollary 14.1.

Assume ${\hat{Z}}\subset\Omega_{k}$ is compact and shift invariant. The following are equivalent

(1)

${\hat{Z}}$ is kfsm
(2)

For $j=1,\dots,k$ , ${\hat{\ell}}_{j}({\hat{Z}})\leq{\hat{r}}_{j}({\hat{Z}})$ with indices reduced mod $2k$

If $Z$ is recurrent we know that each $Z\in{\mathcal{B}}_{k}(f)$ has $\iota_{k}({\hat{Z}})=B_{k}(\omega,\vec{\nu})$ for some allowable $(\omega,\vec{\nu})$ which yields an indirect connection between the interpolated semi-monotone maps and HM parameterization.

There is a well known connection between the dynamics of $d_{n}:z\mapsto z^{n}$ and the full shift on $n$ symbols. This yields a connection of the symbolic kfsm sets as described by this corollary to invariant sets of the circle on which the action of $d_{n}$ is semi-monotone, sometimes call circular orbits.

In Figure 6 we show the conditions forced by Corollary 14.1 as flat spots in the graph of $d_{n}$ for $k=3$ and $n=6$ . There are two classes of flat spots. Those in class A are forced by the condition that ${\hat{Z}}\subset\Omega_{k}$ and thus satisfies (5.2). These are the intervals of width $1/6$ , $[1/18,2/9],[7/18,5/9]$ and $[13/18,8/9]$ . These conditions are satisfied by all symbolic kfsm sets in the corollary. The other three flat spots in class B are determined by the conditions in part (b) of the corollary and vary with the symbolic kfsm set. Note that the result of addding all the flat spots is a degree one semi-monotone circle map as expected. See figures in [20] and [33].

The figure also illustrates a clear difference between the kfsm sets for bimodal circle maps and circular orbits for $d_{n}$ -one. Specifically, the kfsm sets correspond to a specific subclass of circular orbits for $d_{2k}$ . On the other hand, there is clearly a tight relationship between the theories which needs to be investigated. Perhaps the degree reduction process described in [7, 33] would be a good place to start.

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On the abundance of kk-fold semi-monotone minimal sets in bimodal circle maps

Abstract.

1. introduction

Theorem 1.1.

Theorem 1.2.

2. Preliminaries

2.1. Dynamics

2.2. The circle, finite covers and degree one circle maps

Theorem 2.1.

Proof.

Remark 2.2.

2.3. Rotation number and interval

Remark 2.3.

3. Semi-monotone degree-one maps

3.1. Definition and basic properties

Lemma 3.1.

Definition 3.2.

Remark 3.3.

3.2. Finitely many flat spots

Definition 3.4.

Lemma 3.5.

Proof.

4. A class of bimodal circle maps and their positive slope orbits

4.1. The class 𝒢{\mathcal{G}}

Definition 4.1.

Remark 4.2.

4.2. the model map

4.3. Positive slope orbits

5. Symbolic description of positive slope orbits

5.1. the itinerary maps

5.2. Symbolic analogs of covering spaces

Definition 5.1.

Lemma 5.2.

Lemma 5.3.

Proof.

Remark 5.4.

5.3. Rotation numbers and sets

6. Topological conjugacies and the image of the itinerary maps

Theorem 6.1.

Proof.

Remark 6.2.

7. kk-fold semi-monotone sets

7.1. Definitions

Definition 7.1.

7.2. interpolation

Lemma 7.2.

Definition 7.3.

Remark 7.4.

7.3. symbolic k-fold semi-monotone sets and the map gg

Definition 7.5.

Theorem 7.6.

Proof.

Definition 7.7.

Lemma 7.8.

Proof.

7.4. Rotation numbers and sets

Lemma 7.9.

Proof.

Definition 7.10.

Remark 7.11.

8. The HM construction

8.1. definition and basic properties

Definition 8.1.

Remark 8.2.

8.2. Cyclic covers

Definition 8.3.

Remark 8.4.

Theorem 8.5.

Proof.

Remark 8.6.

9. Parameterization of ℬk​(g){\mathcal{B}}_{k}(g) and 𝒩k​(g){\mathcal{N}}_{k}(g) by the HM construction

9.1. Resonance and holes

Definition 9.1.

Remark 9.2.

Lemma 9.3.

Proof.

9.2. continuity and injectivity

Remark 9.4.

Theorem 9.5.

Proof.

On the abundance of $k$ -fold semi-monotone minimal sets in bimodal circle maps

4.1. The class ${\mathcal{G}}$

7. $k$ -fold semi-monotone sets

7.3. symbolic k-fold semi-monotone sets and the map $g$

9. Parameterization of ${\mathcal{B}}_{k}(g)$ and ${\mathcal{N}}_{k}(g)$ by the HM construction

10. kfsm sets in $S^{1}$ and $\Omega_{1}$

10.1. In $S^{1}$

10.2. Symbolic kfsm sets in $\Omega_{1}$

10.5. Sturmian minimal sets, the case $k=1$

11. Structure of $\operatorname{HM}_{k}(g)$

12.1. periods in $\Omega_{1}$

12.3. The pure lattice and the structure of $\operatorname{HM}_{kp/q}$

12.5. Estimating the number of $p/q$ kfsm sets

13.1. The family of k-fold interpolated maps for $g\in{\mathcal{G}}$