Quasi-self-similar fractals containing “Y” have dimension larger than one

Let $(X,d_{X})$ and $(Z,d_{Z})$ be compact metric spaces. A homeomorphism $h:X\to Z$ is a quasi-symmetry if there is a homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that for every $x,y,z\in X$ with $x\neq z$, we have

One can think of quasi-symmetries as homeomorphisms that do not distort shapes much.

We say that two metric spaces $X$ and $Z$ are quasi-symmetrically equivalent, or simply quasi-symmetric, and write $X\sim_{\mathrm{qs}}Z$, if there is a quasi-symmetric homeomorphism between $X$ and $Z$. Quasi-symmetries define an equivalence relation on the collection of metric spaces.

We often consider different metrics on the same topological space $M$. In this case, we say that two metrics $d_{1}$ and $d_{2}$ on $M$ are quasi-symmetric if the identity map $\mathrm{id}:(M,d_{1})\to(M,d_{2})$ is a quasi-symmetry.

For a compact metric space $X$, the conformal dimension of $X$, denoted by $\operatorname{Cdim}(X)$, is defined as the infimal Hausdorff dimension in the quasi-symmetric equivalence class containing $X$, i.e.,

We say that $X$ attains its conformal dimension if the infimum in the definition of the conformal dimension of $X$ is realized as the minimum.

We abbreviate coarse expanding conformal as “cxc”. The idea of cxc maps were introduced by P. Haïssinsky and K. Pilgrim in [HP09]. We give precise definitions in Section 2.

We say that a topological dynamical system $f:X\to X$ on a topological space $X$ is topological cxc if it satisfies 3 properties (Expansion), (Degree), and (Irreducibility): (Expansion) and (Irreducibility) are standard conditions for topological dynamical systems. The property (Degree) prohibits periodic and more generally recurrent periodic branch points, which characterizes semi-hyperbolic rational maps in complex dynamics. We remark that we do not use a metric to define topological cxc maps.

Now let us assume that $X$ is equipped with a metric $d_{X}$. We say that $f:X\to X$ is metric cxc if (1) $f$ is topological cxc, and (2) $f$ satisfies metric conditions defined in such a way that the Sullivan’s Principle of the Conformal Elevator holds. It guarantees that the metric space $(X,d_{X})$ is quasi-self-similar in a sense that any arbitrarily small piece of $X$ looks similar to a large piece of $X$ up to some bounded error. More precisely, a metric space $X$ is called quasi-self-similar if there is a $k\geq 1$ and $r_{0}>0$ such that for any ball $B\subset X$ with radius $r<r_{0}$ there exists a map $E_{B}:B\to X$ satisfying

for every $x,y\in B$ [Sul82, p. 42].

Although the definition of cxc maps takes motivation in complex dynamics, cxc maps also include self-maps of manifolds whose iterates are uniformly quasi-regular. Being topological or metric cxc is preserved by quasi-symmetric conjugacies and taking products. See [HP12] for more examples of cxc maps.

Let $f:X\to X$ be a topological cxc map. If $f$ is metric cxc for two different metrics $d$ and $d^{\prime}$ on $X$, then $(X,d)$ and $(X,d^{\prime})$ are quasi-symmetric. See Proposition 2.4. Hence, there exists a unique quasi-symmetric class $\mathcal{G}(f)$, called the canonical gauge of $f$, consisting of metrics for which $f$ is metric cxc. In this sense, the conformal dimension for $\mathcal{G}(f)$, i.e., the infimal Hausdorff dimension of metrics in $\mathcal{G}(f)$, is a natural invariant for topological cxc maps.

For a topological cxc map $f:X\to X$, we can construct metrics, called visual metrics, which are contained in $\mathcal{G}(f)$. One advantage of having visual metrics is the following: Equipping $X$ with a visual metric, we may assume that the $n$-th preimages of a ball are quasi-balls with uniformly bounded distortion whose radii decay exponentially fast about $n>0$.

J. Azzam proved that if a compact connected metric space $X$ is antenna-like, i.e., having “sufficiently many” copies of $\mathbb{Y}$ with uniformly bounded distortion, then $\operatorname{Hdim}(X)>1$ [Azz15]. See Section 2.3 for precise definitions.

Since being antenna-like is preserved by quasi-symmetries, to show Theorem A it suffices to prove that $X$ with a visual metric is antenna-like. Starting from one copy of $\mathbb{Y}$ in $X$, we first find many other copies of $\mathbb{Y}$ by using properties of topological cxc maps, and then we show these copies are not distorted much by using the properties of metric cxc maps.

In complex dynamics with one variable, we consider rational maps $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$, which are holomorphic maps from the Riemann sphere $\hat{\mathbb{C}}$ to itself. In this article, we discuss three sub-families of rational maps. For a rational map $f$,

• •

  $f$ is semi-hyperbolic if $f$ does not have any parabolic periodic point nor any recurrent critical point;

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  $f$ is sub-hyperbolic if any critical point is periodic, preperiodic, or attracted to an attracting periodic cycle; and

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  $f$ is post-critically finite if any critical point is periodic or preperiodic.

Post-critically finiteness implies sub-hyperbolicity, and sub-hyperbolicity implies semi-hyperbolicity. Every Fatou component of a semi-hyperbolic rational map is in an attracting or a super-attracting basin [Mañ93]. Any sub-hyperbolic rational map can be obtained from a unique post-critically finite rational map by quasi-conformal deformation on the Fatou set, which deforms each super-attracting basin to an attracting basin [McM88].

For applications of conformal dimensions to geometric group theory or complex dynamics, we use a slight variant of the conformal dimension, called the Ahlfors-regular conformal dimension. A metric space $X$ is Ahlfors-regular if round sets of $X$ well behave with the $\delta$-Hausdorff measure $\mathcal{H}^{\delta}$ where $\delta=\operatorname{Hdim}(X)$, in a sense that there exists $C>1$ such that for any $0<r<\frac{1}{2}\cdot\operatorname{diam}(X)$ and $x\in X$ we have

For a compact metric space $X$, the Ahlfors-regular conformal dimension, denoted by $\operatorname{ARConfdim}(X)$, is defined by

M. Bonk and B. Kleiner showed that the attainment of conformal dimensions characterizes lattices of hyperbolic isometry groups.

A similar result for self-maps of the 2-sphere was established by P. Haïssinsky and K. Pilgrim.

A semi-hyperbolic rational map $f$ acts on its Julia set $\mathcal{J}_{f}$ as a metric cxc system. If the semi-hyperbolic rational map is a polynomial and the Julia set $\mathcal{J}_{f}$ is connected, then the conformal dimension of $\mathcal{J}_{f}$ equals 1, see [CP12], [Kin17].

The degree-$d$ negated Chebyshev polynomials are (-1) times the degree-$d$ Chebyshev polynomials.

Since $\operatorname{ARConfdim}\geq\operatorname{Cdim}$ in general, Theorem C holds even if we change the conformal dimension to Ahlfors-regular conformal dimension. Theorem C may be considered as a complex dynamical analogue of Theorem 1.1 for one dimension.

Julia sets of semi-hyperbolic quadratic polynomials are related to fractals generated by classical iterated function systems [ERS10]. For instance, the Julia set of $z^{2}+i$ is shown to be quasi-symmetric equivalent to a universal object called the continuum self-similar tree, see [BM20] and [BM22]. The following is then an immediate corollary of Theorem A.

The authors are grateful to Kevin Pilgrim and Dylan Thurston for useful conversation.

The first author was supported by the Simons Foundation Institute Grant Award ID 507536 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI.

In this subsection, we assume spaces $X$ and $Z$ are compact, Hausdorff, connected, and locally connected topological spaces, and $f:X\to Z$ is a finite-to-one continuous map. We define the degree of $f$ by

and the local degree of $f$ at $x\in X$ by

where the infimum is taken over all neighborhoods $U$ of $x$.

The following properties of fbc’s are shown in [HP09, Lemma 2.1.2 and Lemma 2.1.3]: For an fbc $f:X\to Z$,

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  $f$ is open, closed, onto, and proper;

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  the set of branch points $B_{f}:=\{x\in X:\deg(f;x)>1\}$ is nowhere dense in $X$;

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  the set of branch values $V_{f}:=f(B_{f})$ is nowhere dense in $Z$;

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  if $U\subset Z$ is open and connected, then its inverse image $f^{-1}(U)$ is a union of disjoint open subsets $f^{-1}(U)=\widetilde{U}_{1}\sqcup\ldots\sqcup\widetilde{U}_{m}$ where $f:\widetilde{U}_{i}\to U$ is an fbc of degree $d_{i}$, and $d_{1}+\ldots+d_{k}=\deg(f)$.

We refer the reader to [Edm76] and [HP09] for more details on finite branched coverings.

Recall that $o$ denotes the center of the space $\mathbb{Y}$, i.e., the point where the three arms meet.

When we consider a topological cxc system $f:X\to X$, we always assume that $X$ is non-singleton, compact, connected, locally connected, and metrizable.

Let $\mathcal{U}_{0}$ be a cover of $X$ consisting of finitely many open connected subsets. For any $n>0$, we inductively define a cover $\mathcal{U}_{n}$ by the collection of connected components of inverse images of elements of $\mathcal{U}_{n}$ under $f$. We define $\mathbf{U}:=\bigcup_{n\geq 0}\mathcal{U}_{n}$ the collection of all open sets in the covers $\mathcal{U}_{n}$’s. If $U\in\mathcal{U}_{n}$, then we say that the level of $U$ is $n$ and write $|U|=n$.

The dynamical system $f:X\to X$ is topological cxc if there is a finite open cover $\mathcal{U}_{0}$ satisfying the following three properties:

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  (Expansion) $\operatorname{mesh}(\mathcal{U}_{n})\to 0$ as $n\to\infty$. More precisely, for any open cover $\mathcal{V}$ of $X$, there is $N>0$ such that for any $U\in\mathcal{U}_{n}$ with $n>N$ there is $V\in\mathcal{V}$ satisfying $U\subset V$. If $X$ is a metric space, it is equivalent to $\max\{\operatorname{diam}U:U\in\mathcal{U}_{n}\}\to 0$ as $n\to\infty$.

• •

  (Degree)

  $$\max\{\deg(f^{k}:\widetilde{U}\to U):k\in\mathbb{N},~{}\widetilde{U}\in\mathcal{U}_{k},~{}U\in\mathcal{U}_{0}\}<\infty.$$

• •

  (Irreducibility) For any open set $W\subset X$, there exists $n\in N$ with $f^{n}(W)=X$.

For a metric space $(X,d_{X})$, a dynamical system $f:X\to X$ is metric cxc if it is topological cxc about a finite open cover $\mathcal{U}_{0}$ and, in addition, the following conditions hold for the same $\mathcal{U}_{0}$: There exist

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  continuous, increasing embeddings $\rho_{\pm}:[1,\infty)\to[1,\infty)$, called the forward and backward roundness distortion functions, and

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  increasing homeomorphisms $\delta_{\pm}:[0,1]\to[0,1]$, called the forward and backward relative diameter distortion functions,

such that the following two properties hold.

• •

  (Roundness distortion) For all $n,k\in\mathbb{N}$ and for all $U\in\mathcal{U}_{n},~{}\widetilde{U}\in\mathcal{U}_{n+k},~{}\tilde{y}\in\widetilde{U}$, and $y\in U$, if $f^{\circ k}(\widetilde{U})=U$ and $f^{\circ k}(\tilde{y})=y$, then we have the backward roundness bound

  $$\mathrm{round}(\widetilde{U},\tilde{y})<\rho_{-}(\mathrm{round}(U,y)),$$

  and the forward roundness bound

  $$\mathrm{round}(U,y)<\rho_{+}(\mathrm{round}(\widetilde{U},\tilde{y})),$$

  where for an interior point $a$ of $A$ the roundness of $A$ about $a$ is defined as

  $$\mathrm{round}(A,a)=\frac{\sup\{|x-a|:x\in A\}}{\sup\{r:B(a,r)\subset A\}}.$$

• •

  (Diameter distortion) For all $n_{0},n_{1},k\in\mathbb{N}$ and for all $U\in\mathcal{U}_{n_{0}},~{}U^{\prime}\in\mathcal{U}_{n_{1}},~{}\widetilde{U}\in\mathcal{U}_{n_{0}+k}$, and $\widetilde{U}^{\prime}\in\mathcal{U}_{n_{1}+k}$ with $\widetilde{U}^{\prime}\subset\widetilde{U}$ and $U^{\prime}\subset U$, if $f^{k}(\widetilde{U})=U$ and $f^{k}(\widetilde{U}^{\prime})=U^{\prime}$, then

  $$\frac{\operatorname{diam}\widetilde{U}^{\prime}}{\operatorname{diam}\widetilde{U}}<\delta_{-}\left(\frac{\operatorname{diam}U^{\prime}}{\operatorname{diam}U}\right)$$

  and

  $$\frac{\operatorname{diam}U^{\prime}}{\operatorname{diam}U}<\delta_{+}\left(\frac{\operatorname{diam}\widetilde{U}^{\prime}}{\operatorname{diam}\widetilde{U}}\right).$$

From [HP09], we have that the property of being metric cxc is preserved by quasi-symmetric conjugacies.

A dynamical system $f:X\to X$ that is topological cxc with respect to a finite open cover $\mathcal{U}_{0}$ determines a class of so-called visual metrics. Visual metrics are compatible with the dynamics and $\mathcal{U}_{0}$ in a sense that any element in $\bf{U}$ can be thought of as a ball whose size exponentially decays as the level increases. The following results summarize key properties of the visual metrics; see [HP09, Chapter 3].

In fact, we have $d(x,y)=e^{-\varepsilon}\cdot d(f(x),f(y))$ for $x,y\in B$ where $B$ is a sufficiently small ball that does not contain a branch point [HP09, Proposition 3.2.3].

The following proposition implies that a visual metric is a canonical metric, in some sense, with which the topological cxc map becomes metric cxc.

We call the quasi-symmetric equivalence class $\mathcal{G}(f)$ of metrics on $X$ that contains visual metrics the canonical gauge (or the conformal gauge) of $f$. The conformal dimension $\operatorname{Cdim}((X,\mathcal{G}(f)))$ (resp. the Ahlfors-regular conformal dimension $\operatorname{ARConfdim}((X,\mathcal{G}(f)))$) denotes the infimal Hausdorff dimension of metrics (resp. Ahlfors-regular metrics) on $X$ in the canonical gauge $\mathcal{G}(f)$.

We refer the reader to [Azz15] for a detailed account of this subsection.

The following theorem is a combination of the results in [Azz15].

For $d\geq 1$, the degree-$d$ Chebyshev polynomial $T_{d}$ is defined by $T_{d}(\cos{\theta})=\cos(d\cdot\theta)$. The Julia set of $T_{d}$ is a closed interval $[-1,1]$ so that $T_{d}:[-1,1]\to[-1,1]$ is well-defined. We always have $T_{d}(1)=1$ but $T_{d}(-1)=-1$ if $d$ is odd and $T_{d}(-1)=1$ if $d$ is even.

We have

and

so that $\deg(f;x_{i})=\deg(f;y_{i})=2$ for any $i,j$. By reordering the indices, we have

Then $f$ homeomorphically maps each connected component of $[-1,1]\setminus f^{-1}(\{-1,1\})$ to $(-1,1)$.

We have

and

so that $\deg(f;x_{i})=\deg(f;y_{j})=2$ for any $i,j$. By reordering the indices, we have

Again $f$ homeomorphically maps each connected component of $[-1,1]\setminus f^{-1}(\{-1,1\})$ to $(-1,1)$.

Now we consider the negated Chebyshev polynomial $T^{\prime}_{d}:=-T_{d}$. If $d=2n$ then $T^{\prime}_{d}$ and $T_{d}$ are conjugate each other by the negation $z\mapsto-z$. Suppose $d=2n+1$. Since $T^{\prime}_{d}$ swaps the two end points $\{-1,1\}$, $T^{\prime}_{d}$ is not conjugate to $T_{d}$. With the notations used in Case (i) above, we have $f^{-1}(-1)=\{1,y_{1},y_{2},\dots,y_{n}\}$ and $f^{-1}(1)=\{-1,x_{1},x_{2},\dots,x_{n}\}$.

One can construct the degree-$d$ Chebyshev polynomial $T_{d}:[-1,1]\to[-1,1]$ by projecting the map $z\mapsto z^{d}$ on the unit circle $\mathbb{S}^{1}$ onto the diameter $[-1,1]$ on the real axis. Suppose $d=2n+1$. The map $\exp(i\theta)\mapsto\exp(i(\pi-\theta))$ is the reflection of $\mathbb{S}^{1}$ in the imaginary axis. It follows from

that the projection of the map $z\mapsto z^{d}$ on $\mathbb{S}^{1}$ onto the diameter on the imaginary axis $[-i,i]$ is well-defined. This vertical projection gives rises to the negated Chebyshev polynomial dynamics.

Suppose $f:X\to X$ is metric cxc with respect to $\mathcal{U}_{0}$ and $X$ is equipped with properties (1)-(4) in Proposition 2.3. We also use the constants $C,r_{0}$, and $\varepsilon$ for used in Proposition 2.3. Suppose that there is an embedding $\mathbb{Y}\hookrightarrow X$.

By Theorem 2.5, it suffices to show that $X$ is $c^{\prime\prime}$-antenna-like for some $c^{\prime\prime}>0$. We divide the proof into three steps.

• (1)

  We first show in Lemma 4.1 that there is $c>0$ so that every $U\in\mathcal{U}_{0}$ has a $c$-antenna.

• (2)

  Then we can use Lemma 4.2 to show that there exists $c^{\prime}>0$ so that every $U\in\mathbf{U}$ has a $c^{\prime}$-antenna.

• (3)

  Then Lemma 4.3 applies to show that $X$ is $c^{\prime\prime}$-antenna-like for some $c^{\prime\prime}>0$.

Let us show the three lemmas.