Soft bounds on the Hausdorff dimension of dynamical Cantor sets

Posted by: matheuscmss | February 4, 2017

Soft bounds on the Hausdorff dimension of dynamical Cantor sets

Many problems in several areas of Mathematics (including Dynamical Systems and Number Theory) can “reduced” to the analysis of dynamical Cantor sets: for instance, the theorems of Newhouse, Palis and Takens on homoclinic bifurcations of surfaces diffeomorphisms, and the theorems of Hall, Freiman and Moreira on the structure of the classical Lagrange and Markov spectra rely on the study of dynamical Cantor sets of the real line.

The wide range of applicability of dynamical Cantor sets is partly explained by the fact that several natural examples of Cantor sets are defined in terms of Dynamical Systems: for example, Cantor’s ternary set ${C}$ is

$\displaystyle C=\bigcap\limits_{n\in\mathbb{N}} T^{-n}([0,1])$

where ${T:[0,1/3]\cup [2/3,1]\rightarrow [0,1]}$ is ${T(x)=3x (\textrm{ mod } 1)}$ .

In some applications of dynamical Cantor sets, it is important to dispose of estimates on their Hausdorff dimensions: for instance, the celebrated work of Bourgain and Kontorovich on Zaremba’s conjecture needs particular types of dynamical Cantor sets with Hausdorff dimension close to one.

For this reason, a considerable literature on this topic was developed. Among the diverse settings covered by many authors, one finds the articles of Bumby, Hensley, …, Jenkinson-Pollicott, Falk-Nussbaum, where the so-called thermodynamical methods are exploited to produce approximations for the Hausdorff dimension of Cantor sets defined in terms of continued fraction expansions (i.e., Cantor sets of number-theoretical nature).

In general, the thermodynamical methods quoted above provide a sequence ${s_n}$ of fast-converging approximations for the Hausdorff dimension ${HD(K)}$ of dynamical Cantor sets: for instance, the algorithm described by Jenkinson-Pollicott here gives a sequence ${s_n}$ converging to ${HD(E_2)}$ at super-exponential speed, i.e., ${|s_n-HD(E_2)|\leq C \theta^{n^2}}$ for some constants ${C>0}$ and ${0<\theta<1}$ , where ${E_2}$ is the Cantor set of real numbers whose continued fraction expansions contain only ${1}$ and ${2}$ .

In particular, the thermodynamical methods give good heuristics for the first several digits of the Hausdorff dimension of dynamical Cantor sets (e.g., if we list ${s_n}$ for ${1\leq n\leq 10}$ and the first three digits of ${s_n}$ coincide for all ${5\leq n\leq 10}$ , then it is likely that one has found the first three digits of ${HD(K)}$ ).

The heuristic bounds provided by the thermodynamical methods can be turned into rigorous estimates: indeed, one of the goals of the recent work of Jenkinson-Pollicott consists into rigorously computing the first 100 digits of ${HD(E_2)}$ .

However, the conversion of heuristic bounds into rigorous estimates is not always easy, and, for this reason, sometimes a slowly converging method producing two sequences ${\alpha_n}$ and ${\beta_n}$ of rigorous bounds (i.e., ${\alpha_n<HD(K)<\beta_n}$ for all ${n\in\mathbb{N}}$ ) might be interesting for practical purposes.

In this post, we explain a method described in pages 68 to 70 of Palis-Takens book giving explicit sequences ${\alpha_n\leq HD(K)\leq \beta_n}$ converging slowly (e.g., ${\beta_n-\alpha_n\leq C/n}$ for some constant ${C>0}$ and all ${n\in\mathbb{N}}$ ) towards ${HD(K)}$ , and, for the sake of comparison, we apply it to exhibit crude bounds on the Hausdorff dimensions of some Cantor sets defined in terms of continued fraction expansions.

1. Dynamical Cantor sets of the real line

Definition 1 A ${C^m}$ -dynamical Cantor set ${K\subset\mathbb{R}}$ is

$\displaystyle K=\bigcap\limits_{n\in\mathbb{N}}\Psi^{-n}(I_1\cup\dots\cup I_r)$

where:

${\Psi:I_1\cup\dots\cup I_r\rightarrow I}$ is an expanding ${C^m}$ -map (i.e., ${|\Psi'(x)|>1}$ for every ${x}$ ) from a finite union ${I_1\cup\dots\cup I_r}$ of pairwise disjoint closed intervals to the convex hull ${I}$ of ${I_1\cup\dots\cup I_r}$ ;

${I_1,\dots, I_r}$ is a Markov partition, that is, ${\Psi(I_j)}$ is the convex hull of the union of some of the intervals ${I_1,\dots, I_r}$ , and

${\Psi|_K}$ is topologically mixing, i.e., for some ${n_0\in\mathbb{N}}$ , ${\Psi^n(K\cap I_j)=K}$ for all ${1\leq j\leq r}$ .

Example 1 As we already mentioned, Cantor’s ternary set ${C}$ is a dynamical Cantor set:

$\displaystyle C=\bigcap\limits_{n\in\mathbb{N}} T^{-n}([0,1])$

where ${T:[0,1/3]\cup [2/3,1]\rightarrow [0,1]}$ is the affine map ${T(x)=3x (\textrm{ mod } 1)}$ .

Example 2 Let ${B=\{\beta_1,\dots,\beta_r\}}$ be a finite alphabet of finite words ${\beta_j\in(\mathbb{N}^*)^{r_j}}$ . The Cantor set

$\displaystyle K(B) = \{[0;\gamma_1,\gamma_2,\dots]:\gamma_i\in B \, \, \forall \, i\in\mathbb{N}\}$

of real numbers ${\alpha}$ whose continued fraction expansions ${\alpha=[0;a_1,a_2,\dots]:=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}}$ are given by concatenations of the words in ${B}$ is a dynamical Cantor set.In fact, it is possible to construct intervals ${I(\beta_j)}$ such that

$\displaystyle K(B) = \bigcap\limits_{n\in\mathbb{N}}\Psi^{-n}(I(\beta_1)\cup\dots\cup I(\beta_r))$

where ${\Psi|_{I(\beta_j)} = G^{r_j}}$ and ${G(x)=\{1/x\}}$ is the Gauss map: see, e.g., this paper here for more explanations.

By definition, a dynamical Cantor set ${K}$ can be inductively constructed as follows. We start with the Markov partition ${\mathcal{R}_1 = \{I_1,\dots,I_r\}}$ given by the connected components of the domain of the expanding map ${\Psi}$ defining ${K}$ . For each ${n\geq 2}$ , we define ${\mathcal{R}_n}$ as the collection of connected components of ${\Psi^{-1}(J)}$ , ${J\in\mathcal{R}_{n-1}}$ .

For later use, for each ${R\in\mathcal{R}_n}$ , we denote by

$\displaystyle \lambda_{n,R} := \inf |(\Psi^n)'|_{R}| \quad \textrm{and} \quad \Lambda_{n,R} := \sup |(\Psi^n)'|_{R}|$

1.1. Hausdorff dimension and box counting dimension

Recall that the Hausdorff dimension ${HD(A)}$ and the box-counting dimension ${d(A)}$ of a compact set ${A\subset\mathbb{R}}$ are defined as follows.

The Hausdorff ${s}$ -measure ${m_s(A)}$ of ${A}$ is

$\displaystyle m_s(A)=\lim\limits_{\delta\rightarrow 0}\inf\left\{\sum\limits_{a=1}^c \textrm{diam}(U_a)^s: (U_a)_{a=1}^c \textrm{ finite open cover of } A \textrm{ of diameter } <\delta\right\}$

and the Hausdorff dimension of ${A}$ is

$\displaystyle HD(A) = \inf\{s\in [0,1]: m_s(A)=0\} = \sup\{s\in[0,1]: m_s(A)=\infty\}$

The box-counting dimension is

$\displaystyle d(A) = \lim\limits_{\varepsilon\rightarrow 0} \frac{\log N_{\varepsilon}(A)}{\log(1/\varepsilon)}$

where ${N_{\varepsilon}(A)}$ is the smallest number of intervals of lengths ${\leq\varepsilon}$ needed to cover ${A}$ .

Exercise 1 Show that ${HD(A)\leq d(A)}$ .

1.2. Upper bound on the dimension of dynamical Cantor sets

Let ${K}$ be a ${C^1}$ -dynamical Cantor set associated to an expanding map ${\Psi:I_1\cup\dots\cup I_r\rightarrow I}$ . Consider the quantities ${\beta_n}$ defined by

$\displaystyle \sum\limits_{R\in\mathcal{R}_n} \left(\frac{1}{\lambda_{n,R}}\right)^{\beta_n}=1$

Proposition 2 For all ${n\geq 1}$ , one has ${d(K)\leq\beta_n}$ .

Proof: Let ${n\geq 1}$ and fix ${\beta>d(K)}$ . By definition, there exists ${\varepsilon_0>0}$ such that

$\displaystyle N_{\varepsilon}(K)\leq 1/\varepsilon^{\beta}$

for all ${0<\varepsilon\leq\varepsilon_0}$ .

In other words, given ${0<\varepsilon\leq\varepsilon_0}$ , we can cover ${K}$ using a collection of intervals ${\{J_1, \dots, J_m\}}$ with ${m\leq 1/\varepsilon^{\beta}}$ such that every ${J_l}$ has length ${\leq\varepsilon}$ .

It follows from the definitions that, for each ${R\in\mathcal{R}_n}$ , the pre-images of the intervals ${J_l}$ under ${\Psi^n|_R}$ form a covering of ${K\cap R}$ by intervals of length ${\leq \varepsilon/\lambda_{n,R}}$ . Therefore,

$\displaystyle N_{\varepsilon/\lambda_{n,R}}(K\cap R)\leq 1/\varepsilon^{\beta}$

for all ${R\in\mathcal{R}_{n}}$ and ${0<\varepsilon\leq\varepsilon_0}$ , and, a fortiori,

$\displaystyle N_{\delta}(K\cap R)\leq 1/(\lambda_{n,R}\delta)^{\beta}$

for all ${R\in\mathcal{R}_n}$ and ${0<\delta\leq\varepsilon_0/\lambda_{n,R}}$ .

Hence, if we define ${\lambda_n=\sup\limits_{R\in\mathcal{R}_n} \lambda_{n,R}}$ , then

$\displaystyle N_{\delta}(K)\leq \frac{1}{\delta^{\beta}}\left(\sum\limits_{R\in\mathcal{R}_n}\frac{1}{\lambda_{n,R}^{\beta}}\right)$

for all ${0<\delta\leq \varepsilon_0/\lambda_n}$ .

By iterating this argument ${k}$ times, we conclude that

$\displaystyle N_{\delta}(K)\leq \frac{1}{\delta^{\beta}}\left(\sum\limits_{R\in\mathcal{R}_n}\frac{1}{\lambda_{n,R}^{\beta}}\right)^k$

for all ${0<\delta\leq \varepsilon_0/\lambda_n^k}$ .

Thus,

$\displaystyle d(K) \leq \beta + \lim\limits_{k\rightarrow\infty}\frac{\log\left(\sum\limits_{R\in\mathcal{R}_n}\frac{1}{\lambda_{n,R}^{\beta}}\right)^k}{\log(\lambda_n^k/\varepsilon_0)} = \beta + \frac{\log\left(\sum\limits_{R\in\mathcal{R}_n}\frac{1}{\lambda_{n,R}^{\beta}}\right)}{\log\lambda_n}$

Since ${\beta>d(K)}$ is arbitrary, we deduce from the previous inequality that

$\displaystyle 0\leq \frac{\log\left(\sum\limits_{R\in\mathcal{R}_n}\left(\frac{1}{\lambda_{n,R}}\right)^{d(K)}\right)}{\log\lambda_n}$

Because ${\lambda_n>1}$ , we get that ${\sum\limits_{R\in\mathcal{R}_n}\left(\frac{1}{\lambda_{n,R}}\right)^{d(K)}\geq 1}$ , and, a fortiori, ${d(K)\leq \beta_n}$ . $\Box$

1.3. Lower bound on the dimension of dynamical Cantor sets

Let ${K}$ be a ${C^1}$ -dynamical Cantor set associated to an expanding map ${\Psi:I_1\cup\dots\cup I_r\rightarrow I}$ . Consider the sequence ${\beta_n}$ defined in the previous subsection and fix ${\beta_{\infty}\in [0,1]}$ be a constant such that ${\beta_{\infty}\geq\beta_n}$ for all ${n\geq 1}$ (e.g., ${\beta_{\infty}=1}$ certainly works).

Take ${n_0\in\mathbb{N}}$ such that ${\Psi^{n_0+1}(K\cap I_j)=K}$ for all ${1\leq j\leq r}$ and set

$\displaystyle C:=\sup|(\Psi^{n_0})'|^{\beta_{\infty}}\geq 1$

Remark 1 If ${\Psi}$ is a full Markov map, i.e., ${\Psi(K\cap I_j)=K}$ for all ${1\leq j\leq r}$ , then we can choose ${n_0=1}$ and ${C=1}$ .

Consider the quantities ${\alpha_n}$ given by

$\displaystyle \sum\limits_{R\in\mathcal{R}_n} \left(\frac{1}{\Lambda_{n,R}}\right)^{\alpha_n}=C$

Proposition 3 For all ${n\geq 1}$ , one has ${\alpha_n\leq HD(K)}$ .

Proof: Suppose by contradiction that ${HD(K)<\alpha_n}$ and take ${HD(K)<\alpha<\alpha_n}$ .

By definition, ${m_{\alpha}(K)=0}$ , so that for every ${\varepsilon>0}$ there is a finite cover ${(U^{(\varepsilon)}_a)_{a=1}^{c(\varepsilon)}}$ of ${K}$ with

$\displaystyle \sum\limits_{a=1}^{c(\varepsilon)} \textrm{diam}(U_a)^\alpha\leq\varepsilon$

Note that any interval of length strictly smaller than

$\displaystyle \kappa_0:=\min\{dist(I, J): I, J\in\mathcal{R}_n, I\neq J\}/2 > 0$

intersects at most one ${R\in\mathcal{R}_n}$ .

Thus, if we define ${\varepsilon_0:=\kappa_0^{1/\alpha}}$ , then each element of the cover ${\mathcal{U}:=(U^{(\varepsilon_0)}_a)_{a=1}^{c(\varepsilon_0)}}$ of ${K}$ intersects at most one element of ${\mathcal{R}_n}$ . Hence, if we define

$\displaystyle \mathcal{U}_R:=\{U\in\mathcal{U}: U\cap R\neq\emptyset\},$

then, given any ${R\in\mathcal{R}_n}$ , one has that ${\mathcal{U}_R}$ has fewer elements than ${\mathcal{U}}$ .

Consider ${n_0\in\mathbb{N}}$ such that ${\Psi^{n_0+1}(K\cap R)=K}$ for all ${R\in\mathcal{R}_1}$ . From the definitions, for each ${R\in\mathcal{R}_n}$ , we see that ${(\Psi^{n+n_0}|R)(\mathcal{U}_R)}$ is a well-defined cover of ${K}$ such that

$\displaystyle \sum\limits_{V\in(\Psi^{n+n_0}|_R)(\mathcal{U}_R)} \textrm{diam}(V)^\alpha\leq (\sup|(\Psi^k)'|)^{\alpha} \cdot\Lambda_{n,R}^{\alpha} \cdot\sum\limits_{U\in\mathcal{U}_R}\textrm{diam}(U)^\alpha$

Since ${\alpha<\alpha_n<\beta_n\leq\beta_{\infty}}$ , we get that

$\displaystyle \sum\limits_{V\in(\Psi^{n+n_0}|_R)(\mathcal{U}_R)} \textrm{diam}(V)^\alpha\leq C \cdot\Lambda_{n,R}^{\alpha} \cdot\sum\limits_{U\in\mathcal{U}_R}\textrm{diam}(U)^\alpha$

We want to exploit this estimate to prove that

$\displaystyle \sum\limits_{V\in(\Psi^{n+n_0}|_{R_0})(\mathcal{U}_{R_0})} \textrm{diam}(V)^\alpha< \varepsilon_0$

for some ${R_0\in\mathcal{R}_n}$ . In this direction, suppose that

$\displaystyle \sum\limits_{V\in(\Psi^{n+n_0}|_{R})(\mathcal{U}_{R})} \textrm{diam}(V)^\alpha\geq \varepsilon_0 \quad \forall\,\,R\in\mathcal{R}_n$

In this case, the discussion above would imply

$\displaystyle \begin{array}{rcl} \sum\limits_{U\in\mathcal{U}} \textrm{diam}(U)^\alpha &=&\sum\limits_{R\in\mathcal{R}_n}\sum\limits_{U\in\mathcal{U}_R} \textrm{diam}(U)^\alpha \\ &\geq& \sum\limits_{R\in\mathcal{R}_n} (C\cdot\Lambda_{n,R}^{\alpha})^{-1} \left(\sum\limits_{V\in(\Psi^{n+n_0}|_{R})(\mathcal{U}_{R})} \textrm{diam}(V)^\alpha\right) \\ &\geq& C^{-1}\left(\sum\limits_{R\in\mathcal{R}_n}\Lambda_{n,R}^{-\alpha}\right) \varepsilon_0, \end{array}$

a contradiction because on one hand ${\sum\limits_{U\in\mathcal{U}} \textrm{diam}(U)^\alpha\leq \varepsilon_0}$ (by definition of ${\mathcal{U}}$ ) and on the other hand ${\left(\sum\limits_{R\in\mathcal{R}_n}\Lambda_{n,R}^{-\alpha}\right)>C}$ (by our choice of ${\alpha<\alpha_n}$ ).

In summary, we assumed that ${HD(K)<\alpha_n}$ , we considered an arbitrary cover ${\mathcal{U}}$ of ${K}$ with

$\displaystyle \sum\limits_{U\in\mathcal{U}} \textrm{diam}(U)^\alpha\leq \varepsilon$

and we found a cover ${\mathcal{V}:=(\Psi^{n+n_0}|_{R_0})(\mathcal{U}_{R_0})}$ of ${K}$ with fewer elements than ${\mathcal{U}}$ such that

$\displaystyle \sum\limits_{V\in(\Psi^{n+n_0}|_{R_0})(\mathcal{U}_{R_0})} \textrm{diam}(V)^\alpha< \varepsilon_0$

By iterating this argument, we would end up with a cover of ${K}$ containing no elements, a contradiction. This proves that ${\alpha_n\leq HD(K)}$ . $\Box$

1.4. Slow convergence of towards the Hausdorff dimension

Let ${K}$ be a ${C^2}$ -dynamical Cantor set associated to an expanding map ${\Psi:I_1\cup\dots\cup I_r\rightarrow I}$ . In general, the sequences ${\alpha_n\leq HD(K)\leq \beta_n}$ discussed above converge slowly towards ${HD(K)}$ :

Proposition 4 One has ${\beta_n-\alpha_n=O(1/n)}$ .

Proof: The so-called bounded distortion property (see Theorem 1 in Chapter 4 of Palis-Takens book or this blog post here) ensures the existence of a constant ${a=a(K)\geq 1}$ such that

$\displaystyle \Lambda_{n,R}\leq a\lambda_{n,R}$

for all ${n\in\mathbb{N}}$ and ${R\in\mathcal{R}_n}$ .

Let ${\lambda:=\inf|\Psi'|>1}$ and, for all ${n\geq \log a/\log\lambda}$ , define

$\displaystyle \delta_n:=\frac{\alpha_n\log a + \log C}{-\log a + n\log\lambda}$

In this setting, we have that

$\displaystyle \begin{array}{rcl} \sum\limits_{R\in\mathcal{R}_n}\left(\frac{1}{\lambda_{n,R}}\right)^{\alpha_n+\delta_n} &\leq& a^{\alpha_n+\delta_n} \sum\limits_{R\in\mathcal{R}_n}\left(\frac{1}{\Lambda_{n,R}}\right)^{\alpha_n+\delta_n} \leq \frac{a^{\alpha_n+\delta_n}}{\lambda^{n\delta_n}} \sum\limits_{R\in\mathcal{R}_n}\left(\frac{1}{\Lambda_{n,R}}\right)^{\alpha_n} \\ &=& \frac{a^{\alpha_n+\delta_n}}{\lambda^{n\delta_n}} C = 1 \end{array}$

Therefore, ${\beta_n\leq \alpha_n+\delta_n}$ , that is,

$\displaystyle \beta_n-\alpha_n\leq \frac{\alpha_n\log a + \log C}{-\log a + n\log\lambda} \leq \frac{HD(K)\log a + \log C}{-\log a + n\log\lambda} = O(1/n)$

This proves the proposition. $\Box$

2. Hausdorff dimension of Gauss-Cantor sets

In this section, we apply the previous discussion in the context of Gauss-Cantor sets ${K(B)}$ , that is, the dynamical Cantor sets from Example 2 above.

For the sake of simplicity of exposition, we will not describe the calculation of the sequences ${\alpha_n}$ and ${\beta_n}$ approaching ${HD(K(B))}$ for a general ${B}$ , but we shall focus on two particular examples.

2.1. Some bounds on the Hausdorff dimension of ${E_2}$

Consider the alphabet ${B=\{1,2\}}$ consisting of the words ${1}$ and ${2}$ . The corresponding Cantor set ${K(\{1,2\})}$ is the Cantor set — denoted by ${E_2}$ in the beginning of this post — of real numbers whose continued fraction expansions contain only ${1}$ and ${2}$ .

The theory of continued fractions (see Moreira’s paper and Cusick-Flahive book) says that ${E_2}$ is the dynamical Cantor set associated to the restriction ${\Psi}$ of the Gauss map ${G(x)=\{1/x\}}$ to ${I_1\cup I_2}$ , where

$\displaystyle I_2=[[0;2,\overline{1,2}],[0;2,\overline{2,1}]] \quad \textrm{and} \quad I_1 = [[0;1,\overline{1,2}], [0;1,\overline{2,1}]]$

Note that the functions ${\Phi_1(x)=\frac{1}{1+x}}$ and ${\Phi_2(x)=\frac{1}{2+x}}$ defined on the interval ${I=[[0;\overline{2,1}],[0;\overline{1,2}]]}$ are the inverse branches of ${G:I_1\cup I_2\rightarrow I}$ .

Remark 2 Here, ${\overline{a_1,\dots,a_n}}$ denotes the infinite word obtained by periodic repetition of the block ${a_1,\dots,a_n}$ .

By applying ${\Psi^{-(n-1)}}$ to the Markov partition ${\mathcal{R}_1=\{I_2, I_1\}}$ , we deduce that ${\mathcal{R}_n}$ consists of the intervals ${I(a_1,\dots,a_n)}$ , ${(a_1,\dots,a_n)\in\{1,2\}^n}$ , with extremities

$\displaystyle [0;a_1,\dots,a_n,\overline{1,2}] \quad \textrm{ and } \quad [0;a_1,\dots,a_n,\overline{2,1}]$

The quantities ${\lambda_{n,R}}$ and ${\Lambda_{n,R}}$ are not hard to compute using the following remarks. First, ${(\Psi^n)'|_R}$ is monotone on each ${R\in\mathcal{R}_n}$ (because ${(\Psi^n)|_R}$ are Möbius transformations induced by integral matrices with determinant ${\pm1}$ ), so that the values ${\inf|(\Psi^n)'|_R|}$ and ${\sup|(\Psi^n)'|_R|}$ are attained at the extremities of ${R}$ . Secondly, the derivative of the Gauss map is ${G'(x)=-1/x^2}$ . By combining these facts, we get that

$\displaystyle \lambda_{n,R} = \min\left\{\prod\limits_{j=0}^{n-1}\frac{1}{[0;a_{j+1},\dots,a_n,\overline{1,2}]^2}, \prod\limits_{j=0}^{n-1}\frac{1}{[0;a_{j+1},\dots,a_n,\overline{2,1}]^2}\right\}$

and

$\displaystyle \Lambda_{n,R} = \max\left\{\prod\limits_{j=0}^{n-1}\frac{1}{[0;a_{j+1},\dots,a_n,\overline{1,2}]^2}, \prod\limits_{j=0}^{n-1}\frac{1}{[0;a_{j+1},\dots,a_n,\overline{2,1}]^2}\right\}$

Hence, the sequences ${\alpha_n}$ and ${\beta_n}$ defined in the previous section are the solutions of the equations

$\displaystyle \sum\limits_{(a_1,\dots,a_n)\in\{1,2\}^n}\max\left\{\prod\limits_{j=0}^{n-1}[0;a_{j+1},\dots,a_n,\overline{1,2}], \prod\limits_{j=0}^{n-1}[0;a_{j+1},\dots,a_n,\overline{2,1}]\right\}^{2\beta_n} = 1$

and

$\displaystyle \sum\limits_{(a_1,\dots,a_n)\in\{1,2\}^n}\min\left\{\prod\limits_{j=0}^{n-1}[0;a_{j+1},\dots,a_n,\overline{1,2}], \prod\limits_{j=0}^{n-1}[0;a_{j+1},\dots,a_n,\overline{2,1}]\right\}^{2\alpha_n} = 1$

(Here, we used Remark 1 and the fact that ${G:I_1\cup I_2\rightarrow I}$ is a full Markov map in order to get the equation of ${\alpha_n}$ .)

Of course, these equations allow to find the first few terms of the sequences ${(\alpha_n)_{n\in\mathbb{N}}}$ and ${(\beta_n)_{n\in\mathbb{N}}}$ approaching ${HD(E_2)}$ with some computer-aid: for example, this Mathematica routine here shows that

$\displaystyle \alpha_2 = 0.504748..., \quad \dots \quad \alpha_5=0.519149..., \quad \dots \quad \alpha_{12}=0.526178...$

and

$\displaystyle \beta_2 = 0.563479..., \quad \dots \quad \beta_5=0.545092..., \quad \dots \quad \beta_{12}=0.536916...$

In particular, ${0.526178...=\alpha_{12}\leq HD(E_2)\leq\beta_{12}=0.536916...}$

Remark 3 The approximations ${\alpha_{12}}$ and ${\beta_{12}}$ of ${HD(E_2)}$ (obtained from computing with the words in ${\{1,2\}^{12}}$ ) are very poor in comparison with the approximation

$\displaystyle s_{12}=0.531 280 506 277 205 141 592 435 787 861 805 662 052 800 294 6593...$

provided by the thermodynamical method of Jenkinson-Pollicott (also based on calculations with the words in ${\{1,2\}^{12}}$ ). In fact, Jenkinson-Pollicott showed in this paper here that the first 100 digits of ${HD(E_2)}$ are

$\displaystyle \begin{array}{rcl} HD(E_2)&=&0.53128050627720514162446864736847178549305910901839 \\ & & 87798883978039275295356438313459181095701811852398... \end{array}$

Thus, the first 18 digits of ${s_{12}}$ accurately describe ${HD(E_2)}$ (while only the first two digits of ${\beta_{12}}$ and the first digit of ${\alpha_{12}}$ are accurate).

2.2. Some bounds on the Hausdorff dimension of ${K(\{1,2_2\})}$

Let us consider now the alphabet ${B}$ consisting of the words ${1}$ and ${2,2:=2_2}$ . The theory of continued fractions says that the convex hull of ${K(\{1,2_2\})}$ is the interval ${I}$ with extremities ${[0;\overline{2}]}$ and ${[0;1,\overline{2}]}$ . The images ${I_1:=\phi_1(I)}$ and ${I_{2,2} := \phi_{2,2}(I)}$ of ${I}$ under the inverse branches

$\displaystyle \phi_1(x):=\frac{1}{1+\frac{1}{x}} \quad \textrm{and} \quad \phi_{2,2}(x) := \frac{1}{2+\frac{1}{2+\frac{1}{x}}}$

of the first two iterates of the Gauss map ${G(x)=\{1/x\}}$ provide the first step of the construction of the Cantor set ${K(\{1,2_2\})}$ . In general, given ${n\in\mathbb{N}}$ , the collection ${\mathcal{R}^n}$ of intervals of the ${n}$ th step of the construction of ${K(\{1,2_2\})}$ is given by

$\displaystyle \mathcal{R}_n:=\{\phi_{x_1}\circ\dots\circ\phi_{x_n}(I): (x_1,\dots, x_n)\in\{1, 2_2\}^n\}$

Hence, the interval ${R=\psi_{x_1}\circ\dots\circ\psi_{x_n}(I)\in\mathcal{R}^n}$ associated to a string ${(x_1,\dots, x_n)\in\{0,1\}^n}$ has extremities ${[0;x_1,\dots,x_n,\overline{2}]}$ and ${[0;x_1,\dots,x_n,1,\overline{2}]}$ .

Similarly as in the previous subsection, we conclude that the quantities ${\lambda_{n,R}}$ and ${\Lambda_{n,R}}$ are given by

$\displaystyle \lambda_{n,R} = \min\left\{\prod\limits_{i=1}^{n}\left(\frac{1}{[0;x_i,\dots, x_n, \overline{2}]}\right)^2, \prod\limits_{i=1}^{n}\left(\frac{1}{[0;x_i,\dots, x_n, 1, \overline{2}]}\right)^2 \right\}$

and

$\displaystyle \Lambda_{n,R} = \max\left\{\prod\limits_{i=1}^{n}\left(\frac{1}{[0;x_i,\dots, x_n, \overline{2}]}\right)^2, \prod\limits_{i=1}^{n}\left(\frac{1}{[0;x_i,\dots, x_n, 1, \overline{2}]}\right)^2 \right\}$

Thus, ${\alpha_n}$ and ${\beta_n}$ are the solutions of

$\displaystyle \sum\limits_{(x_1,\dots,x_n)\in\{1,22\}^n}\left(\min\{[0;x_i,\dots, x_n, \overline{2}], [0;x_i,\dots, x_n, 1, \overline{2}]\}\right)^{2\alpha_n}=1$

and

$\displaystyle \sum\limits_{(x_1,\dots,x_n)\in\{1,22\}^n}\left(\max\{[0;x_i,\dots, x_n, \overline{2}], [0;x_i,\dots, x_n, 1, \overline{2}]\}\right)^{2\beta_n}=1$

Once more, we can calculate the first terms of the sequences ${(\alpha_n)_{n\in\mathbb{N}}}$ and ${(\beta_n)_{n\in\mathbb{N}}}$ with some computer-aid: this Mathematica routine here reveals that

$\displaystyle \alpha_2 = 0.344871..., \quad \dots \quad \alpha_7=0.352094..., \quad \dots \quad \alpha_{12}=0.353465...$

and

$\displaystyle \beta_2 = 0.370012..., \quad \dots \quad \beta_5=0.359745..., \quad \dots \quad \beta_{12}=0.357917...$

In particular, ${0.353465...=\alpha_{12}\leq HD(E_2)\leq\beta_{12}=0.357917...}$

Remark 4 After implementing of the Jenkinson-Pollicott method with this Mathematica routine here, we get the following approximations for ${HD(K(\{1, 2_2\}))}$ :

$\displaystyle s_2=0.383019..., \quad s_4=0.355052..., \quad \dots s_{10}=0.35540048..., \quad s_{12}=0.355398...$

The super-exponential convergence of Jenkinson-Pollicott method suggests that

$\displaystyle HD(K(\{1, 2_2\})) = 0.355...$

Note that the first two digits of this approximation are accurate (because the first two digits of ${\alpha_{12}}$ and ${\beta_{12}}$ coincide).

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Disquisitiones Mathematicae

Soft bounds on the Hausdorff dimension of dynamical Cantor sets

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