Families of infinite parabolic IFS with overlaps: the approximating method

This work should be understood in the background of families of overlapping iterated function systems. A typical family of overlapping iterated function system (IFS) on a compact metric space consists of a flow of finitely many strictly contractive endomorphisms on the space parametrized by some (time) parameter. The dimension of attractors and measures supported on them (especially the invariant or ergodic ones) is the focus in the theory of families of IFS. Besides calculating the dimension of attractors and measures supported on the attractors, another important problem is to decide whether the projective measures on the attractors from measures on the symbolic spaces are singular or absolutely continuous with respect to the Lebesgue measure, refer to [BRS, Hoc1, Hoc3, PSS, Shm2, Shm3, SS1, SSS]. See also [Fur] by Furstenberg, [Hoc4] by Hochman, [PS1] by Peres-Solomyak and [Shm4] by Shmerkin for more interesting questions on families of overlapping IFS. These problems are already very difficult in the case of Bernoulli convolutions, see for example [BV1, BV2, Hoc1, LPS, Shm1, Sol3, SS2, Var1, Var2, Var3] for recent progress on the topic.

Note that most of the above research are done on families of finite IFS, that is, every individual IFS in the family is constituted by finitely many maps. In this work we try to deal with families of infinite IFS. We choose the families of parabolic iterated function systems investigated by K. Simon, B. Solomyak and M. Urbański in [SSU1, SSU2] to demonstrate our ideas, which are probably applicable to some other families of infinite IFS. Our technique here is to utilize the sequence of concentrating measures on the attractors of families of finite sub-systems to approximate the measures on the attractors of families of infinite systems.

Let $X\subset\mathbb{R}$ be a closed interval with its Borel $\sigma$-algebra $\mathcal{B}$. Let $\mathcal{\hat{M}}(X)$ be the collection of all the finite Borel measures on $(X,\mathcal{B})$. For a set $A\subset\mathbb{R}^{d}$ with $d\in\mathbb{N}$, let $HD(A)$ be its Hausdorff dimension [Fal2].

The two values indicate the distribution of a measure on $X$ from two complementary points of views, with the possibility of the lower one being strictly less than the upper one. However, they coincide with each other for ergodic measures with respect to transformations of proper regularity (for example the $C^{1}$ diffeomorphisms) on $X$. See for example [HS, SimS2, You] on calculation of the Hausdorff dimension of measures in various circumstances. Following P. Mattila, M. Morán and J. M. Rey [MMR], the author discovers some semi-continuity of the measure-dimension mappings $dim_{*}$ and $dim^{*}$ on $\mathcal{\hat{M}}(X)$ equipped with the setwise topology [FKZ, Las], which is crucial in our approximating process.

Now we gradually introduce the families of parabolic iterated function systems. For $\vartheta\in(0,1]$, a point $v\in X$ is called an indifferent point of a $C^{1+\vartheta}$ map $s:X\rightarrow X$ if

$|s^{\prime}(v)|=1$.

It is called hyperbolic if $0<|s^{\prime}(x)|<1$ for any $x\in X$. Parabolic maps with finitely many indifferent points can be handled in our context, with some technical modifications on the proofs.

The collection of PIFS $S=\{s_{i}:X\rightarrow X\}_{i\in I}$ satisfying

$\cup_{i\in I\setminus\{i_{1}\}}s_{i}(X)\subset X^{o}\setminus\{v\}$

is termed $\Gamma_{X}(\vartheta)$. For an integer $d\in\mathbb{N}$ and an open set $U\subset\mathbb{R}^{d}$, we focus on families of PIFS instead of a single system in $\Gamma_{X}(\vartheta)$ parametrized by the vector-time parameter $\mathbb{t}\in U$. The parabolic map is always fixed at any time in the family. Since families of finite PIFS (that is, $\#I<\infty$) has been considered by Simon-Solomyak-Urbański, we focus on the families of infinite PIFS ($\#I=\infty$) in the following. That is, we focus on families of PIFS

$S^{\mathbb{t}}=\{s_{i}^{\mathbb{t}}:X\rightarrow X\}_{i\in\mathbb{N}}$,

such that $S^{\mathbb{t}}\in\Gamma_{X}(\vartheta)$ with the parabolic map $s_{1}^{\mathbb{t}}=s_{1}$ remaining the same at any time $\mathbb{t}\in U$. Let

$\pi_{\mathbb{t}}:I^{\infty}\rightarrow X$

be the projection map and $J_{\mathbb{t}}=\pi_{\mathbb{t}}(I^{\infty})$ be the attractor ([Hut]) at time $\mathbb{t}\in U$. To achieve some reasonable conclusions on the parametrized family, obviously some control on the dependence of the family with respect to $\mathbb{t}$ is necessary. A family of PIFS $S^{\mathbb{t}}=\{s_{i}^{\mathbb{t}}:X\rightarrow X\}_{i\in\mathbb{N}}$ is said to satisfy the continuity condition if for any fixed $i\in\mathbb{N}$, the map $s_{i}^{\mathbb{t}}$ depends continuously on the parameter $\mathbb{t}\in U$ in the Banach space $C^{1+\vartheta}$ equipped with the supreme norm. Let $\mathfrak{L}^{d}$ be the Lebesgue measure on $\mathbb{R}^{d}$ for $d\in\mathbb{N}$. The family is said to satisfy the transversality condition if there exists a constant $C_{1}$ such that

$\mathfrak{L}^{d}\{\mathbb{t}\in U:|\pi_{\mathbb{t}}(\omega)-\pi_{\mathbb{t}}(\tau)|\leq r\}\leq C_{1}r$

for any $\omega,\tau\in\mathbb{N}^{\infty},\omega_{1}\neq\tau_{1}$ and any $r>0$.

While the continuity condition comes more naturally, the transversality condition has been recognized to be an effective condition to achieve some uniform conclusions in various contexts of flows of dynamical systems. For example, see [Fal1, PolS, PS2, PS3, SimS1, Sol1, Sol2].

The following are some classical notions on the symbolic dynamics on $\mathbb{N}^{\infty}$. For an infinite word $\omega=\omega_{1}\omega_{2}\cdots\in\mathbb{N}^{\infty}$, let

$\sigma(\omega)=\omega_{2}\omega_{3}\cdots$

be the shift map. For a finite $k$-word $\tau\in\mathbb{N}^{k}$, let

$[\tau]=\{\omega\in\mathbb{N}^{\infty}:\omega|_{k}=\tau\}$

be a cylinder set, in which $\omega|_{k}=\omega_{1}\omega_{2}\cdots\omega_{k}$ is the $k$-th restriction of the infinite word $\omega$. Let

$\mathbb{N}^{*}=\cup_{k=1}^{\infty}\mathbb{N}^{k}$

be the collection of all the finite words. Now consider the $\sigma$-algebra $\mathcal{B}_{\mathbb{N}^{\infty}}$ generated by all the cylinder sets on $\mathbb{N}^{\infty}$. Let $h_{\mu}(\sigma)$ be the entropy of the shift map $\sigma$ with respect to the partition of $\mathbb{N}^{\infty}$ by cylinder sets $\{[i]\}_{i=1}^{\infty}$ for a measure $\mu\in\mathcal{\hat{M}}(\mathbb{N}^{\infty})$ ([Hoc2, Wal]).

One can treat $\mu$ as the law of an infinite discrete-time stochastic process

$\mathbb{Y}=\{Y_{i}\}_{i\in\mathbb{N}}$,

in which $Y_{i}$ is a discrete random variable on $\mathbb{N}$ for $i\in\mathbb{N}$ under the natural projection with law $\mu(\mathbb{N}\times\mathbb{N}\times\cdots\times Y_{i}\times\mathbb{N}\times\cdots)$. The sequence of infinite random variables is called independent if the finite variables

$\{Y_{1},Y_{2},\cdots,Y_{n}\}$

are independent for any $n\in\mathbb{N}$, see [Tsi, Definition 3a(1),(b)]. The stochastic process $\mathbb{Y}$ is a Markov process under the hypothesis of independence, refer to [BHP, FKZ, Hai, HL].

In the following we focus on the measures on $J_{\mathbb{t}}$ projected from probability measures on $\mathbb{N}^{\infty}$ through $\pi_{\mathbb{t}}$ for $\mathbb{t}\in U$. For a measure $\mu$ on the symbolic space $\mathbb{N}^{\infty}$, consider its projection under $\pi_{\mathbb{t}}$:

$\nu_{\mathbb{t}}=\mu\circ\pi_{\mathbb{t}}^{-1}$

for any $\mathbb{t}\in U$. We are particularly interested in the cases when $\mu$ is invariant or ergodic with respect to the shift map $\sigma$ on $\mathbb{N}^{\infty}$. If $\mu$ is invariant then $\nu_{\mathbb{t}}$ is of pure type, see for example [JW]. Let

$\lambda^{\mathbb{t}}_{\mu}(\sigma)=-\int_{\mathbb{N}^{\infty}}\log|s_{\omega_{1}}^{\prime}(\pi_{\mathbb{t}}\circ\sigma(\omega))|d\mu(\omega)$

be the average Lyapunov exponent of the family of PIFS $S^{\mathbb{t}}=\{s_{i}^{\mathbb{t}}:X\rightarrow X\}_{i\in\mathbb{N}}$ at time $\mathbb{t}\in U$.

Our first main result deals with the two basic questions regarding the projective measures $\{\nu_{\mathbb{t}}\}_{\mathbb{t}\in U}$, these are, deciding their dimensions $dim_{*}\nu_{\mathbb{t}},dim^{*}\nu_{\mathbb{t}}$ and whether they are absolutely continuous or singular with respect to the Lebesgue measure for $\mathbb{t}\in U$. Let $\mathbb{N}_{n}=\{1,2,\cdots,n\}$ be the $n$-th truncation of $\mathbb{N}$ for $n\in\mathbb{N}$. We leave the technical concept-the $n$-th concentrating measure $\mu_{n}$ (supported on $\mathbb{N}_{n}^{\infty}$) of a finite measure $\mu$ (supported on $\mathbb{N}^{\infty}$) to Section 4. The following result generalizes [SSU1, Theorem 2.3] from families of finite PIFS to families of infinite PIFS.

Note that the proof of [SSU1, Theorem 2.3] does not apply to Theorem 1.4. There are two main obstacles on application of Simon-Solomyak-Urbański’s techniques to the families of infinite PIFS. The first one is that $h_{\mu}(\sigma)$ can explode in case of $\#I=\infty$, see [Hoc2]. If $h_{\mu}(\sigma)=\infty$ for some ergodic $\mu\in\mathcal{\hat{M}}(\mathbb{N}^{\infty})$ with respect to the shift map, the Shannon-McMillan-Breiman Theorem is not known to hold in this case [Hay]. The second one is the regularity of the families of infinite PIFS. The collection of PIFS $S=\{s_{i}:X\rightarrow X\}_{i\in I}$ satisfying the following conditions is called $\Gamma_{X}(\vartheta,V,\gamma,u,M)$ in [SSU1].

1. (a)

   There exists an open connected neighbourhood $V$ of $v$, such that

   $\cup_{i\in I\setminus\{i_{1}\}}s_{i}(X)\cap V=\emptyset$.

2. (b)

   There exists some $\gamma\in(0,1)$, such that

   $\sup_{i\in I\setminus\{i_{1}\}}\{\|s_{i}^{\prime}\|\}\leq\gamma$.

3. (c)

   There exists some $u\in(0,1)$, such that

   $\inf_{i\in I,x\in X}\{|s_{i}^{\prime}(x)|\}\geq u$.

4. (d)

   There exists $M>0$ such that

   $\|S^{\prime}\|_{\vartheta}=\sup_{i\in I}\sup_{x,y\in X}\Big{\{}\cfrac{|s_{i}^{\prime}(x)-s_{i}^{\prime}(y)|}{|x-y|^{\vartheta}}\Big{\}}\leq M$.

For a finite PIFS, $S\in\Gamma_{X}(\vartheta)$ is enough to guarantee $S\in\Gamma_{X}(\vartheta,V,\gamma,u,M)$ for some parameters $V,\gamma,u,M$. However, this is not true for an infinite PIFS. Our technique of approximating measures on the infinite symbolic space by the sequence of concentrating measures on its finite subspaces successfully overcomes these obstacles.

Our next main result deals with local dimension of the exceptional parameters of a family of infinite PIFS $\{S^{\mathbb{t}}\}_{\mathbb{t}\in U}$. Let $\mu$ be a measure on the symbolic space $\mathbb{N}^{\infty}$ with its concentrating measures $\{\mu_{n}\}_{n\in\mathbb{N}}$. For a subset $G\subset U$ and $0<\alpha<1$, in case the sequence $\Big{\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)}\Big{\}}_{n=1}^{\infty}$ admits a limit for any $\mathbb{t}\in G$, let

$E_{\alpha,G}:=\Big{\{}\mathbb{t}\in G:dim^{*}\nu_{\mathbb{t}}<\lim_{n\rightarrow\infty}\min\big{\{}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},\alpha\big{\}}\Big{\}}$

be the level-$\alpha$ exceptional parameters in $G$. Let

$K_{\alpha,G}=\min\Big{\{}\sup_{\mathbb{t}\in G}\lim_{n\rightarrow\infty}\cfrac{h_{\mu_{n}}(\sigma)}{\lambda^{\mathbb{t}}_{\mu_{n}}(\sigma)},\alpha\Big{\}}+d-1$.

For a parameter set $A\subset U\subset\mathbb{R}^{d}$, let $N_{r}(A)$ be the least number of open balls of radius $r>0$ needed to cover the set $A$. A family of PIFS $S^{\mathbb{t}}=\{s_{i}^{\mathbb{t}}:X\rightarrow X\}_{i\in\mathbb{N}}$ is said to satisfy the strong transversality condition if there exists a constant $C_{2}>0$ such that

$N_{r}\{\mathbb{t}\in U:|\pi_{\mathbb{t}}(\omega)-\pi_{\mathbb{t}}(\tau)|\leq r\}\leq C_{2}r^{1-d}$

for any $\omega,\tau\in\mathbb{N}^{\infty},\omega_{1}\neq\tau_{1}$ and any $r>0$. Obviously this condition is stronger than the transversality condition. We have the following estimation on the upper bound of the Hausdorff dimension of the local level-$\alpha$ exceptional set.

This result is a generalization of [SSU1, Theorem 5.3] from families of finite PIFS to families of infinite PIFS. See also [Kau, Theorem].

Since we are considering families of IFS in this work, it would be very helpful if the measure-dimension mappings

$dim_{*}:\mathcal{\hat{M}}(X)\rightarrow[0,\infty)$

or

$dim^{*}:\mathcal{\hat{M}}(X)\rightarrow[0,\infty)$

admits some continuity property. To discuss the continuity problem, some appropriate topology on $\mathcal{\hat{M}}(X)$ is needed. Since the measures $\{\nu_{\mathbb{t}}\}_{\mathbb{t}\in U}\subset\mathcal{\hat{M}}(X)$ in Theorem 1.4 inherit the parametrization naturally from parametrization of the PIFS, one may try to endow the Euclidean metric on $\{\nu_{\mathbb{t}}\}_{\mathbb{t}\in U}$ through the parameter $\mathbb{t}\in U\subset\mathbb{R}^{d}$ to discuss the continuity of the measure-dimension mappings. The measure-dimension mappings are not always continuous under the Euclidean metric on $\{\nu_{\mathbb{t}}\}_{\mathbb{t}\in U}$ in general consideration, although lower semi-continuity of them can be expected. One can see from the family of Bernoulli convolutions $\{\nu_{\lambda}\}_{\lambda}\subset\mathcal{\hat{M}}([0,1])$ with the contraction ratio $\lambda\in(0,1)$. The measure-dimension mappings

$dim_{*}=dim^{*}:(0,1)\rightarrow[0,1]$

are not always continuous since there are dimension drops from $1$ at the inverses of the Pisot parameters. However, they are lower semi-continuous [HS, Theorem 1.8].

Since we are dealing with measures originated from families of the infinite systems and their finite sub-systems simultaneously in this work, we will abandon the Euclidean metric on $\{\nu_{\mathbb{t}}\}_{\mathbb{t}\in U}\subset\mathcal{\hat{M}}(X)$ to discuss the continuity problem. A well-known topology on $\mathcal{\hat{M}}(X)$ is the weak topology.

Denote the convergence in this sense by $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$. However, The measure-dimension mappings $dim^{*}$ and $dim_{*}$ do not admit any semi-continuity under the weak topology on $\mathcal{\hat{M}}(X)$ [Ma, Theorem 3.1]. It turns out that the setwise topology is the most ideal topology for us to discuss the continuity of the measure-dimension mappings.

One is recommended to refer to [Doo, FKZ, GR, HL, Las, LY] for more equivalent descriptions of the setwise topology on $\mathcal{\hat{M}}(X)$ and its various applications. Denote the convergence in this sense by $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$. The measure-dimension mappings $dim^{*}$ and $dim_{*}$ are lower semi-continuous and upper semi-continuous under the setwise topology respectively on $\mathcal{\hat{M}}(X)$.

Considering generalizations of measures, Theorem 2.3 is still true on the space of finitely additive measures (a finitely additive measure $\nu$ on $(X,\mathcal{B})$ is a function from $\mathcal{B}$ to $[0,\infty)$ satisfying finite additivity instead of $\sigma$-additivity), but not true for a signed measure on $(X,\mathcal{B})$ under the Definition 1.1. Obviously a new meterage for the dimension of signed measures is needed. Let $\mathcal{\hat{M}}_{s}(X)$ be the collection of all the finite signed Borel measures on $X$.

Under this modified definition the measure-dimension mappings $dim_{s}$ and $dim^{s}$ from $\mathcal{\hat{M}}_{s}(X)$ to $[0,\infty)$ still bear some continuity.

In this section we discuss the relationship between absolute continuity of a sequence of measures and absolute continuity of its limit measure (under the weak or setwise topology, in case the sequence converges). We start from the following basic result.

Due to Proposition 3.1, we make the following definition formally.

In fact Proposition 3.1 holds for any topological ambient space $X$. However, it is possible that a sequence of absolutely continuous measures converges weakly to a measure which is not absolutely continuous, as one can see from the Example 3.3. Let $\mathfrak{L}^{d}|_{A}$ be the restriction of $\mathfrak{L}^{d}$ on a set $A\subset\mathbb{R}^{d}$.

Unfortunately, the inverse of Proposition 3.1 is not always true, as one can see from the following example. Let $\delta_{\{x\}}$ be the Dirac probability measure at the point $x\in X$.

It is easy to justify that $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\mathfrak{L}^{1}|_{X}$ as $n\rightarrow\infty$ in Example 3.4. However, there are no measure absolutely continuous with respect to $\mathfrak{L}^{1}|_{X}$ at all in the sequence $\{\nu_{n}\}_{n=1}^{\infty}$. One can see that in Example 3.4 the measures in the sequence are all of mixed type. In fact this is the only reason which prevents the inverse of Proposition 3.1 to be true.

Proposition 3.5 also holds for any topological ambient space $X$, but it is not true for convergent sequences of measures under the weak topology on $\mathcal{\hat{M}}(X)$, as one can also see from our Example 3.3. The above results show that the absolute continuity of convergent sequences of measures and their limit measures are equivalent to each other under the setwise topology on $\mathcal{\hat{M}}(X)$, while this relationship is not true under the weak topology. This is another advantage that the setwise topology takes over the weak topology on $\mathcal{\hat{M}}(X)$.

In this section we do the approximation of a probability measure $\mu$ on the infinite symbolic space $\mathbb{N}^{\infty}$ by the sequence of its concentrating measures $\{\mu_{n}\}_{n=1}^{\infty}$ supported on $\{\mathbb{N}_{n}^{\infty}\}_{n=1}^{\infty}$ respectively. We will prove that some properties are inherited by the concentrating measures from $\mu$, under the hypothesis that the sequence of infinite random variables in the infinite stochastic process

$\mathbb{Y}=\{Y_{i}\}_{i\in\mathbb{N}}$

is independent under the law $\mu$. As indicated in Remark 1.6, one may be able to remove the independent hypothesis in due course if one tries the approximation of $\mu$ by its restrictions $\{\mu|_{\mathbb{N}_{n}^{\infty}}\}_{n=1}^{\infty}\subset\mathcal{\hat{M}}(\mathbb{N}_{n}^{\infty})$ instead of the concentrating measures $\{\mu_{n}\}_{n=1}^{\infty}\subset\mathcal{M}(\mathbb{N}_{n}^{\infty})$.

Since for any $q$-level cylinder $[i_{1}i_{2}\cdots i_{q}]\subset\mathbb{N}_{n}^{\infty}$ with $i_{1},i_{2},\cdots i_{q}\in\mathbb{N}_{n}$ and $q\in\mathbb{N}$, the conditions in Definition 4.1 (1)-(3) imply

$\mu_{n}([i_{1}i_{2}\cdots i_{q}])=\sum_{j=1}^{n}\mu_{n}([i_{1}i_{2}\cdots i_{q}j])$

between cylinders of successive levels, the existence and uniqueness of the measure $\mu_{n}$ are guaranteed by the Kolmogorov extension theorem, see for example [Tao, Theorem 2.4.3]. Moreover, $\mu_{n}$ is invariant for any $n\in\mathbb{N}$ if $\mu$ is invariant with respect to the shift map $\sigma$ according to [Ber, Theorem 2]. In fact, more properties are inherited from $\mu$ as the structure of the original measure is suitably preserved considering the structure of the concentrating measures $\{\mu_{n}\}_{n\in\mathbb{N}}$ on the cylinder sets, under the independent hypothesis of $\mathbb{Y}$ with respect to $\mu$.