Hausdorff dimension of frequency sets in beta-expansions

TL;DR

This paper proves a key relationship between the Hausdorff dimensions of sets in beta-expansions and their projections, simplifies the calculation of these dimensions for frequency sets, and provides exact formulas for pseudo-golden ratios.

Contribution

It offers a rigorous proof of a folklore result, simplifies the computation of Hausdorff dimensions for frequency sets, and derives explicit formulas for multinacci numbers.

Findings

Hausdorff dimension of sets in shift space equals that of their projections

Dimension calculation reduces to optimizing finitely many variables

Exact formulas obtained for frequency sets of multinacci numbers

Abstract

By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $\beta>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_\beta$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$. It has been used in some former papers without proof. Then we clarify that for calculating the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit order when the $\beta$-expansion of $1$ is finite. Concretely, it suffices to optimize a function with finitely many variables under some restrictions. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $\beta$'s, which are called pseudo-golden ratios (also called multinacci numbers).

Equations

$$x=\sum _{n=1}^{\infty }\frac{w_n}{{\beta }^n}$$

$$d_{\beta }(w,v):={\beta }^{-\inf \{k\ge 0:w_{k+1}\ne v_{k+1}\}}\text{for }w=(w_n{)}_{n\ge 1},v=(v_n{)}_{n\ge 1}\in {\mathcal{A}}_{\beta }^{\mathbb{N}}.$$

$${\pi }_{\beta }(w):=\frac{w_1}{\beta }+\frac{w_2}{{\beta }^2}+\cdots +\frac{w_n}{{\beta }^n}+\cdots \text{for }w=(w_n{)}_{n\ge 1}\in S_{\beta }.$$

$${\dim }_H(Z,d_{\beta })={\dim }_H{\pi }_{\beta }(Z).$$

$$F_{\beta,a}:=\Big{\{}x\in[0,1):\lim_{n\to\infty}\frac{\#\{1\leq k\leq n:\varepsilon_{k}(x,\beta)=0\}}{n}=a\Big{\}},$$

$${\dim }_H{F}_{2,a}=\frac{-a\log a-(1-a)\log (1-a)}{\log 2}\text{for all }a\in [0,1].$$

$${\dim }_H{F}_{\beta ,a}=\frac{a\log a-(2a-1)\log (2a-1)-(1-a)\log (1-a)}{\log \beta }$$

$$[w]:=\{v\in S_{\beta }:v\text{ begins with }w\}$$

$$\sigma (w_1{w}_2\cdots ):=w_2{w}_3\cdots \text{for }w\in {\mathcal{A}}_{\beta }^{\mathbb{N}}.$$

$$\dim_{H}F_{\beta,a}=\frac{1}{\log\beta}\cdot\max\Big{\{}h_{\mu}(\sigma):\mu\in\mathcal{M}_{\sigma}(S_{\beta}),\mu[0]=a,\mu\text{ is an }(m-1)\text{-Markov measure}\Big{\}}.$$

$$\dim_{H}F_{\beta,a}=\frac{1}{\log\beta}\cdot\max\Big{\{}\mathfrak{h}_{\mu}(\beta,m):\mu\text{ is an }(\beta,m,a)\text{-coordinated set function}\Big{\}},$$

$${\mathfrak{h}}_{\mu }(\beta ,m):=-\sum _{w_1\cdots w_m\in {\Sigma }_{\beta }^m}\mu [w_1\cdots w_m]\log \frac{\mu [w_1\cdots w_m]}{\mu [w_1\cdots w_{m-1}]},$$

$$\mu [0]=a,\mu [1]=1-a,\sum _{\begin{array}{c}v=0,1\\ wv\in {\Sigma }_{\beta }^{\ast }\end{array}}\mu [wv]=\mu [w]\text{and}\sum _{\begin{array}{c}u=0,1\\ uw\in {\Sigma }_{\beta }^{\ast }\end{array}}\mu [uw]=\mu [w]$$

$${\dim }_H{F}_{\beta ,a}=\frac{1}{\log \beta }\cdot \underset{x_1,\cdots ,x_{m-2}}{\max }{f}_a(x_1,\cdots ,x_{m-2})$$

$$=a\log a$$

$$-(x_1-x_2)\log (x_1-x_2)$$

$$\cdots$$

$$-(x_{m-3}-x_{m-2})\log (x_{m-3}-x_{m-2})$$

$$-(1-a-x_1-\cdots -x_{m-2})\log (1-a-x_1-\cdots -x_{m-2})$$

$$-(x_1+\cdots +x_{m-3}+2x_{m-2}+a-1)\log (x_1+\cdots +x_{m-3}+2x_{m-2}+a-1)$$

$$\displaystyle dim_{H}F_{\beta,a}=\frac{1}{\log\beta}\Big{(}a\log a$$

$$-\frac{-2a+3-\sqrt{-8a^2+12a-3}}{6}\log \frac{-2a+3-\sqrt{-8a^2+12a-3}}{6}$$

$$\displaystyle-\frac{-a+\sqrt{-8a^{2}+12a-3}}{3}\log\frac{-a+\sqrt{-8a^{2}+12a-3}}{3}\Big{)}.$$

$$T_{\beta }(x):=\beta x-\lfloor \beta x\rfloor \text{for }x\in [0,1]$$

$${\epsilon }_n(x,\beta ):=\lfloor \beta T_{\beta }^{n-1}(x)\rfloor .$$

$$x=\sum _{n=1}^{\infty }\frac{{\epsilon }_n(x,\beta )}{{\beta }^n}$$

$$\varepsilon^{*}(1,\beta):=\left\{\begin{array}[]{ll}\varepsilon(1,\beta)&\mbox{if }\varepsilon(1,\beta)\mbox{ is infinite}\\ (\varepsilon_{1}(1,\beta)\cdots\varepsilon_{m-1}(1,\beta)(\varepsilon_{m}(1,\beta)-1))^{\infty}&\mbox{if }\varepsilon(1,\beta)\mbox{ is finite with length }m\end{array}\right.$$

$${\Sigma }_{\beta }^{\ast }:=\bigcup _{n=1}^{\infty }{\Sigma }_{\beta }^n.$$

$$w\in {\Sigma }_{\beta }⟺{\sigma }^k(w)\prec {\epsilon }^{\ast }(1,\beta )\text{for all }k\ge 0$$

$$w\in S_{\beta }⟺{\sigma }^k(w)⪯{\epsilon }^{\ast }(1,\beta )\text{for all }k\ge 0$$

$$w_1\cdots w_n\in {\Sigma }_{\beta }^{\ast }\text{if and only if}{w}_1\cdots w_m,w_2\cdots w_{m+1},\cdots ,w_{n-m+1}\cdots w_n\in {\Sigma }_{\beta }^{\ast }.$$

$$w_1\cdots w_m,w_2\cdots w_{m+1},\cdots ,w_{n-m+1}\cdots w_n\in {\Sigma }_{\beta }^{\ast }.$$

$$w_1\cdots w_m,w_2\cdots w_{m+1},\cdots ,w_{n-m+1}\cdots w_n⪯{\epsilon }_1\cdots {\epsilon }_{m-1}({\epsilon }_m-1).$$

$${\sigma }^k(w_1\cdots w_n{0}^{\infty })\prec ({\epsilon }_1\cdots {\epsilon }_{m-1}({\epsilon }_m-1){)}^{\infty }\text{for all }k\ge 0.$$

Full text

1. Introduction

Let $\beta>1$ be a real number. Given $x\in[0,1)$, the most common way to $\beta$-expand $x$ as

[TABLE]

is to use the greedy algorithm, which generates (greedy) $\beta$-expansion. It was introduced by Rényi [24] in 1957 and widely studied in the following decades until now [4, 14, 15, 18, 22, 25, 26]. In some other literature, for examples [1, 2, 3, 8, 20, 27], a $\beta$-expansion of a point $x$ is defined to be a sequence $(w_{n})_{n\geq 1}$ satisfying (1.1). Then a point may have many different $\beta$-expansions including the greedy one. Throughout this paper, we use $\beta$-expansion to denote the greedy one defined by the $\beta$-transformation (see Section 2 for definition).

Let $\Sigma_{\beta}$ be the set of admissible sequences (see Definition 2.1) and $S_{\beta}$ be its (topological) closure in the metric space $(\mathcal{A}_{\beta}^{\mathbb{N}},d_{\beta})$, where $\mathcal{A}_{\beta}$ is the alphabet $\{0,1,\cdots,\lceil\beta\rceil-1\}$, $\lceil\beta\rceil$ is the smallest integer no less than $\beta$, $\mathbb{N}$ is the set of positive integers $1,2,3,\cdots$ and $d_{\beta}$ is the usual metric on $\mathcal{A}_{\beta}^{\mathbb{N}}$ defined by

[TABLE]

Besides, we use $\pi_{\beta}:S_{\beta}\rightarrow[0,1]$ to denote the natural projection map given by

[TABLE]

As the first main result in this paper, the following theorem is a folklore result used in some former papers without explicit proof (for example [30, Section 5]).

Theorem 1.1**.**

Let $\beta>1$. For any $Z\subset S_{\beta}$, the Hausdorff dimension of $Z$ in $(S_{\beta},d_{\beta})$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$, i.e.,

[TABLE]

It is worth to note that $\dim_{H}(Z,d_{\beta})\geq\dim_{H}\pi_{\beta}(Z)$ follows immediately from the fact that $\pi_{\beta}$ is Lipschitz continuous. But even if omitting countable many points to make $\pi_{\beta}$ invertible, the inverse is not Lipschitz continuous. This makes the proof of the inverse inequality much more intricate. We will prove it by using a covering property (see Proposition 2.5) given by Bugeaud and Wang in 2014 deduced from the distribution of full cylinders.

In the following, we consider the digit frequencies of the expansions. This is a classical research topic began by Borel in 1909. His well known normal number theorem [5] implies that, for Lebesgue almost every $x\in[0,1]$, the digit frequency of zeros in its binary expansion is equal to $\frac{1}{2}$. Given $\beta>1$, for any $a\in[0,1]$, let $F_{\beta,a}$ be the set of those $x$’s with digit frequencies of [math]’s equal to $a$ in their $\beta$-expansions. That is the frequency set

[TABLE]

where $\varepsilon_{k}(x,\beta)$ is the $k$th digit in the $\beta$-expansion of $x$ and $\#$ denotes the cardinality. For $\beta=2$, Borel’s normal number theorem means that $F_{2,\frac{1}{2}}$ is of full Lebesgue measure, and implies that $F_{2,a}$ is of zero Lebesgue measure for $a\neq\frac{1}{2}$. This leaves a natural question: How large is $F_{2,a}$ in the sense of dimension? Forty years later, another well known result given by Eggleston [9] showed that

[TABLE]

For the case that $\beta$ is not an integer, the above question, about giving concrete formulae for the Hausdorff dimension of frequency sets, is almost entirely open. Although the Hausdorff dimension of frequency sets can be given by some variational formulae (see for examples [11, 28, 29]), they are abstract and concrete formulae are very scarce. The only known concrete formula is in [17, Theorem 1.2], which contains the well known case that when $\beta=\frac{\sqrt{5}+1}{2}$ is the golden ratio (i.e., the $\beta$-expansion of $1$ is $\varepsilon(1,\beta)=110^{\infty}$), we have

[TABLE]

where $\frac{1}{2}\leq a\leq 1$. See for examples [13, 19]. Note that when $0\leq a<\frac{1}{2}$, $F_{\beta,a}=\varnothing$.

As the second main result in this paper, the next theorem takes a step from abstraction to concreteness. It means that for calculating the Hausdorff dimension of frequency sets, we only need to focus on the entropy (see [31] for definition) with respect to Markov measures of explicit order (see Definition 2.10) when $\beta\in(1,2)$ and the $\beta$-expansion of $1$ is finite. More concretely, it suffices to optimize a function with finitely many variables under some restrictions.

For $\beta\in(1,\infty)$, let $\Sigma_{\beta}^{n}$ be the set of admissible words with length $n\in\mathbb{N}$ and $\Sigma_{\beta}^{*}:=\cup_{n=1}^{\infty}\Sigma_{\beta}^{n}$. For any $w\in\Sigma_{\beta}^{*}$, define

[TABLE]

to be the cylinder in $S_{\beta}$ generated by $w$.

Let $\sigma$ be the shift map on $\mathcal{A}_{\beta}^{\mathbb{N}}$ defined by

[TABLE]

We also use $\sigma$ to denote its restriction on $S_{\beta}$. Let $\mathcal{M}_{\sigma}(S_{\beta})$ be the set of $\sigma$-invariant Borel probability measures on $S_{\beta}$ and $h_{\mu}(\sigma)$ be the measure-theoretic entropy of $\sigma$ with respect to the measure $\mu$. We regard $0\log 0$, $0\log\frac{0}{0}$, $\max\varnothing$ and $\sup\varnothing$ as [math] in the following.

Theorem 1.2**.**

Let $\beta\in(1,2)$ such that $\varepsilon(1,\beta)=\varepsilon_{1}(1,\beta)\cdots\varepsilon_{m}(1,\beta)0^{\infty}$ for some integer $m\geq 2$ with $\varepsilon_{m}(1,\beta)=1$ and let $a\in[0,1]$. Then

[TABLE]

More concretely,

[TABLE]

where for a set function $\mu$ defined from $\{[w]:w\in\cup_{n=1}^{m}\Sigma_{\beta}^{n}\}$ to $[0,1]$,

[TABLE]

and $\mu$ is called $(\beta,m,a)$-coordinated if

[TABLE]

for all $w\in\cup_{n=1}^{m-1}\Sigma_{\beta}^{n}$.

Note that for any $(m-1)$-Markov measure $\mu\in\mathcal{M}_{\sigma}(S_{\beta})$, $h_{\mu}(\sigma)$ is exactly equal to $\mathfrak{h}_{\mu}(\beta,m)$ (see Proposition 2.11).

As an application of the above theorem, in the following we give an exact formula for the Hausdorff dimension of the frequency sets for an important class of $\beta$’s, which are called pseudo-golden ratios.

Theorem 1.3**.**

*Let $\beta\in(1,2)$ such that $\varepsilon(1,\beta)=1^{m}0^{\infty}$ for some integer $m\geq 3$.

(1) If $0\leq a<\frac{1}{m}$, then $F_{\beta,a}=\varnothing$ and $\dim_{H}F_{\beta,a}=0$.

(2) If $\frac{1}{m}\leq a\leq 1$, then*

[TABLE]

where $f_{a}(x_{1},\cdots,x_{m-2})$

[TABLE]

and the maximum is taken over $x_{1},\cdots,x_{m-2}$ such that all terms in the $\log$’s are non-negative. That is, $a\geq x_{1}\geq x_{2}\geq\cdots\geq x_{m-2}\geq 0$ and $x_{1}+\cdots+x_{m-3}+x_{m-2}\leq 1-a\leq x_{1}+\cdots+x_{m-3}+2x_{m-2}$.

In particular, $\dim_{H}F_{\beta,\frac{1}{m}}=\dim_{H}F_{\beta,1}=0$.

Remark 1.4*.*

For the case $m=3$, i.e., $\varepsilon(1,\beta)=1110^{\infty}$, given any $a\in[\frac{1}{3},1]$, by calculating the derivative of $f_{a}(x_{1})$, it is straightforward to get

[TABLE]

In particular, $\dim_{H}F_{\beta,\frac{1}{3}}=\dim_{H}F_{\beta,1}=0$.

We introduce some preliminaries in the next section, and then prove Theorems 1.1, 1.2 and 1.3 in Sections 3, 4 and 5 respectively.

2. Preliminaries

For $\beta>1$, we define the $\beta$-transformation $T_{\beta}:[0,1]\rightarrow[0,1]$ by

[TABLE]

where $\lfloor y\rfloor$ denotes the greatest integer no larger than $y$. For any $n\in\mathbb{N}$ and $x\in[0,1]$, define

[TABLE]

Then we can write

[TABLE]

and we call the sequence $\varepsilon(x,\beta):=\varepsilon_{1}(x,\beta)\varepsilon_{2}(x,\beta)\cdots\varepsilon_{n}(x,\beta)\cdots$ the $\beta$-expansion of $x$. Besides, the sequence $\varepsilon(x,\beta)$ is said to be infinite if there are infinitely many $n\in\mathbb{N}$ such that $\varepsilon_{n}(x,\beta)\neq 0$. Otherwise, there exists a smallest $m\in\mathbb{N}$ such that for any $j>m,\varepsilon_{j}(x,\beta)=0$ but $\varepsilon_{m}(x,\beta)\neq 0$, and we say that $\varepsilon(x,\beta)$ is finite with length $m$.

The quasi-greedy $\beta$-expansion of $1$ defined by

[TABLE]

is very useful for checking the admissibility of a sequence (see Lemma 2.2).

Recall that $\mathcal{A}_{\beta}$ is the alphabet $\{0,1,\cdots,\lceil\beta\rceil-1\}$ and $d_{\beta}$ is the usual metric on $\mathcal{A}_{\beta}^{\mathbb{N}}$.

Definition 2.1** (Admissibility).**

Let $\beta>1$. A sequence $w\in\mathcal{A}_{\beta}^{\mathbb{N}}$ is called admissible if there exists $x\in[0,1)$ such that $\varepsilon_{i}(x,\beta)=w_{i}$ for all $i\in\mathbb{N}$. We denote the set of all admissible sequences by $\Sigma_{\beta}$ and its closure in $(\mathcal{A}_{\beta}^{\mathbb{N}},d_{\beta})$ by $S_{\beta}$. For $n\in\mathbb{N}$, a word $w\in\mathcal{A}_{\beta}^{n}$ is called admissible if there exists $x\in[0,1)$ such that $\varepsilon_{i}(x,\beta)=w_{i}$ for all $i\in\{1,\cdots,n\}$. We denote the set of all admissible words with length $n$ by $\Sigma_{\beta}^{n}$ and write

[TABLE]

One can verify that $\sigma(S_{\beta})=S_{\beta}$. When we write $\sigma^{-1}$, we consider $\sigma$ restricted to $S_{\beta}$. So $\sigma^{-1}A\subset S_{\beta}$ for all $A\subset S_{\beta}$.

The following criterion due to Parry is well known.

Lemma 2.2** ([22]).**

Let $\beta>1$ and $w$ be a sequence in $\mathcal{A}_{\beta}^{\mathbb{N}}$. Then

[TABLE]

and

[TABLE]

where $\prec$ and $\preceq$ denote the lexicographic order in $\mathcal{A}_{\beta}^{\mathbb{N}}$.

We prove the following useful proposition.

Proposition 2.3**.**

Let $\beta>1$ such that $\varepsilon(1,\beta)=\varepsilon_{1}(1,\beta)\cdots\varepsilon_{m}(1,\beta)0^{m}$ for some integer $m\geq 2$ with $\varepsilon_{m}(1,\beta)\neq 0$ and $w_{1}\cdots w_{n}\in\mathcal{A}_{\beta}^{n}$ for some integer $n\geq m$, then

[TABLE]

Proof.

[math] Obvious.

[math] For simplification we use $\varepsilon_{1},\cdots,\varepsilon_{m}$ instead of $\varepsilon_{1}(1,\beta),\cdots,\varepsilon_{m}(1,\beta)$ in the following. Suppose

[TABLE]

By Lemma 2.2 we get

[TABLE]

In order to get $w_{1}\cdots w_{n}\in\Sigma_{\beta}^{*}$, by Lemma 2.2, it suffices to check

[TABLE]

If $k\geq n$, this is obvious. We consider $k\leq n-1$ in the following. Let $l\geq 0$ be the greatest integer such that $k+lm\leq n-1$. Then

[TABLE]

where the last inequality follows from

[TABLE]

which can be proved as follows. In fact, by $w_{n-m+1}\cdots w_{n}\in\Sigma_{\beta}^{*}$ and Lemma 2.2, we get

[TABLE]

This implies (2.1). ∎

In this paper, we use the following definitions of cylinders, noting that in some literature $[w]$ denotes the cylinder in $\Sigma_{\beta}$, not in $S_{\beta}$.

Definition 2.4** (Cylinder).**

Let $\beta>1$. For an admissible word $w\in\Sigma_{\beta}^{*}$ with length $n\in\mathbb{N}$, the cylinder in $S_{\beta}$ of order $n$ generated by $w$ is defined by

[TABLE]

and the cylinder in $[0,1)$ of order $n$ generated by $w$ is defined by

[TABLE]

The following covering property, which plays a crucial role in the proof of Theorem 1.1, is deduced from the length and distribution of full cylinders (see [7, 12, 21] for definition and more details).

Proposition 2.5**.**

([7, Proposition 4.1]) Let $\beta>1$. For any $x\in[0,1)$ and $n\in\mathbb{N}$, the interval $[x-\frac{1}{\beta^{n}},x+\frac{1}{\beta^{n}}]$ intersected with $[0,1)$ can be covered by at most $4(n+1)$ cylinders of order $n$.

Definition 2.6** (Hausdorff measure and dimension in metric space).**

Let $(X,d)$ be a metric space. For any $U\subset X$, denote the diameter of $U$ by $|U|:=\sup_{x,y\in U}d(x,y)$. For any $A\subset X,s\geq 0$ and $\delta>0$, let

[TABLE]

We define the $s$-dimensional Hausdorff measure of $A$ in $(X,d)$ by

[TABLE]

and the Hausdorff dimension of $A$ in $(X,d)$ by

[TABLE]

In the space of real numbers $\mathbb{R}$ (equipped with the usual metric), we use $\mathcal{H}^{s}(A)$ and $\dim_{H}A$ to denote the $s$-dimensional Hausdorff measure and the Hausdorff dimension of $A$ respectively for simplification (see [10]).

Definition 2.7** (Lipschitz continuous).**

Let $(X,d)$ and $(X^{\prime},d^{\prime})$ be two metric spaces. A map $f:X\to X^{\prime}$ is called Lipschitz continuous if there exists a constant $c>0$ such that

[TABLE]

The following basic proposition can be deduced directly from the definitions.

Proposition 2.8**.**

If the map $f:(X,d)\to(X^{\prime},d^{\prime})$ between two metric spaces is Lipschitz continuous, then for any $A\subset X$, we have

[TABLE]

Recall that $\mathcal{M}_{\sigma}(S_{\beta})$ is the set of $\sigma$-invariant Borel probability measures on $S_{\beta}$. The following is a consequence of Carathéodory’s measure extension theorem and the fact that for verifying the $\sigma$-invariance of measures on $S_{\beta}$, one only needs to check it for the cylinders.

Proposition 2.9**.**

Let $\beta\in(1,2]$. Any set function $\mu$ from $\{[w]:w\in\Sigma_{\beta}^{*}\}$ to $[0,1]$ satisfying

[TABLE]

for all $w\in\Sigma_{\beta}^{*}$ can be uniquely extended to be a measure in $\mathcal{M}_{\sigma}(S_{\beta})$.

The following concept is well known (see for examples [11, Section 2] and [16, Section 6.2]).

Definition 2.10** ($k$-Markov measure).**

Let $\beta\in(1,2]$, $k\in\mathbb{N}$ and $\mu\in\mathcal{M}_{\sigma}(S_{\beta})$. We call $\mu$ a $k$-Markov measure if

[TABLE]

for all $w_{1}\cdots w_{n}\in\Sigma_{\beta}^{n}$ with $n>k$.

Recall that $h_{\mu}(\sigma)$ is the measure-theoretic entropy of $\sigma$ with respect to the measure $\mu$. Using $\mathcal{P}:=\{[0],[1]\}$ as a partition generator of the Borel sigma-algebra on $S_{\beta}$, the proof of the following proposition is straightforward.

Proposition 2.11**.**

Let $\beta\in(1,2]$, $k\in\mathbb{N}$ and $\mu\in\mathcal{M}_{\sigma}(S_{\beta})$ be a $k$-Markov measure, then

[TABLE]

3. Proof of Theorem 1.1

The main we need to prove is the following technical lemma.

Lemma 3.1**.**

Let $\beta>1$, $s>0$ and $Z\subset S_{\beta}$. Then for any $\varepsilon\in(0,s)$, we have

[TABLE]

Proof.

Fix $\varepsilon\in(0,s)$. Let $Z_{0}:=Z\cap\Sigma_{\beta}$. Since $S_{\beta}\setminus\Sigma_{\beta}$ is countable, we only need to prove $\mathcal{H}^{s}(Z_{0},d_{\beta})\leq\mathcal{H}^{s-\varepsilon}(\pi_{\beta}(Z_{0}))$.

(1) Choose $\delta_{0}\in(0,\frac{1}{\beta})$ small enough as follows. Since $\beta^{(n+1)\varepsilon}\to\infty$ much faster than $8\beta^{s}n\to\infty$ as $n\to\infty$, there exists $n_{0}\in\mathbb{N}$ such that for any $n>n_{0}$, $8\beta^{s}n\leq\beta^{(n+1)\varepsilon}$. By $\frac{-\log\delta}{\log\beta}-1\to\infty$ as $\delta\to 0^{+}$, there exists $\delta_{0}\in(0,\frac{1}{\beta})$ small enough such that $\frac{-\log\delta_{0}}{\log\beta}-1>n_{0}$. Then for any $n>\frac{-\log\delta_{0}}{\log\beta}-1$, we will have $8\beta^{s}n\leq\beta^{(n+1)\varepsilon}$.

(2) For any $\delta\in(0,\delta_{0})$, let $\{U_{i}\}$ be a $\delta$-cover of $\pi_{\beta}(Z_{0})$, i.e., $0<|U_{i}|\leq\delta$ and $\pi_{\beta}(Z_{0})\subset\cup_{i}U_{i}$. Then for each $U_{i}$, there exists $n_{i}\in\mathbb{N}$ such that $\frac{1}{\beta^{n_{i}+1}}<|U_{i}|\leq\frac{1}{\beta^{n_{i}}}$. By Proposition 2.5, $U_{i}$ can be covered by at most $8n_{i}$ cylinders $I_{i,1},I_{i,2},\cdots,I_{i,8n_{i}}$ of order $n_{i}$. It follows from

[TABLE]

that

[TABLE]

where ($\star$) is because $\frac{1}{\beta^{n_{i}+1}}<|U_{i}|<\delta_{0}$ implies $n_{i}>\frac{-\log\delta_{0}}{\log\beta}-1$, and then by (1) we have $8n_{i}\beta^{s}\leq\beta^{(n_{i}+1)\varepsilon}$. Taking $\inf$ on the right of (3.1), we get $\mathcal{H}^{s}_{\beta\delta}(Z_{0},d_{\beta})\leq\mathcal{H}^{s-\varepsilon}_{\delta}(\pi_{\beta}(Z_{0}))$. It follows from letting $\delta\to 0$ that $\mathcal{H}^{s}(Z_{0},d_{\beta})\leq\mathcal{H}^{s-\varepsilon}(\pi_{\beta}(Z_{0}))$. ∎

Proof of Theorem 1.1.

The inequality $\dim_{H}(Z,d_{\beta})\geq\dim_{H}\pi_{\beta}(Z)$ follows from Proposition 2.8 and the fact that $\pi_{\beta}$ is Lipschitz continuous. The inverse inequality follows from Lemma 3.1. In fact, for any $t<\dim_{H}(Z,d_{\beta})$, there exists $s$ such that $t<s<\dim_{H}(Z,d_{\beta})$. By $\mathcal{H}^{s}(Z,d_{\beta})=\infty$ and Lemma 3.1, we get $\mathcal{H}^{t}(\pi_{\beta}(Z))=\infty$. Thus $t\leq\dim_{H}\pi_{\beta}(Z)$. It means that $\dim_{H}(Z,d_{\beta})\leq\dim_{H}\pi_{\beta}(Z)$. ∎

4. Proof of Theorem 1.2

We will deduce Theorem 1.2 from the following proposition, which is essentially from [23].

Proposition 4.1**.**

Let $\beta>1$ and $a\in[0,1]$. Then

[TABLE]

For the convenience of the readers, we recall some definitions and show how Proposition 4.1 comes from [23].

Definition 4.2**.**

Let $\beta>1$.

(1) For any $w\in S_{\beta}$ and $n\in\mathbb{N}$, the empirical measure is defined by

[TABLE]

where $\delta_{w}$ is the Dirac probability measure concentrated on $w$.

(2) Let $\mathcal{A}$ be an arbitrary non-empty parameter set and let

[TABLE]

where $f_{\alpha}:S_{\beta}\to\mathbb{R}$ is continuous and $c_{\alpha},d_{\alpha}\in\mathbb{R}$ with $c_{\alpha}\leq d_{\alpha}$ for all $\alpha\in\mathcal{A}$. Define

[TABLE]

and

[TABLE]

Combining Theorems 5.2 and 5.3 in [23], we get the following.

Lemma 4.3**.**

Let $\beta>1$. If $\mathcal{M}_{\beta,\mathcal{F}}$ is a non-empty closed connected set, then

[TABLE]

where $h_{top}(S_{\beta,\mathcal{F}},\sigma)$ is the topological entropy of $S_{\beta,\mathcal{F}}$ in the dynamical system $(S_{\beta},d_{\beta},\sigma)$. (See [6] for the definition of the topological entropy for non-compact sets.)

For $\beta>1$ and $a\in[0,1]$, let

[TABLE]

In Definition 4.2 (2), let $\mathcal{F}$ be the singleton $\{(\mathbbm{1}_{[0]},a,a)\}$, where the characteristic function $\mathbbm{1}_{[0]}:S_{\beta}\to\mathbb{R}$ is continuous. (Here we note that another characteristic function $\mathbbm{1}_{[0,\frac{1}{\beta}]}:[0,1]\to\mathbb{R}$ is not continuous, which means that some other similar variational formulae corresponding to dynamical systems on [0,1] can not be applied directly in our case.) We get the following lemma as a special case of the above one.

Lemma 4.4**.**

[TABLE]

Hence, Proposition 4.1 follows from

[TABLE]

where $\pi_{\beta}(S_{\beta,a})\setminus F_{\beta,a}$ is countable since we can check $\pi_{\beta}(S_{\beta,a})\setminus F_{\beta,_{a}}\subset\pi_{\beta}(S_{\beta}\setminus\Sigma_{\beta})$ and Lemma 2.2 implies that $S_{\beta}\setminus\Sigma_{\beta}$ is countable.

Lemma 4.5**.**

([30, Lemma 5.3]) Let $\beta>1$. For any $Z\subset S_{\beta}$, we have

[TABLE]

We give the following proofs to end this section.

Proof of Lemma 4.4.

In Definition 4.2 (2), let $\mathcal{F}$ be the singleton $\{(\mathbbm{1}_{[0]},a,a)\}$. Then

[TABLE]

and

[TABLE]

(1) If $\mathcal{M}_{\beta,a}=\varnothing$, we can prove $S_{\beta,a}=\varnothing$ (and then the conclusion follows).

(By contradiction) If $S_{\beta,a}\neq\varnothing$, there exists $w\in S_{\beta,a}$. For any $n\in\mathbb{N}$, let

[TABLE]

Since $\mathcal{M}(S_{\beta})$ is compact, there exists subsequence $\{\mu_{n_{k}}\}_{k\in\mathbb{N}}\subset\{\mu_{n}\}_{n\in\mathbb{N}}$ and $\mu\in\mathcal{M}(S_{\beta})$ such that $\mu_{n_{k}}\overset{w^{*}}{\to}\mu$ (i.e. $\mu_{n_{k}}$ converge to $\mu$ under the weak* topology). By $\mu_{n_{k}}\circ\sigma^{-1}\overset{w^{*}}{\to}\mu\circ\sigma^{-1}$ and $\mu_{n_{k}}\circ\sigma^{-1}-\mu_{n_{k}}\overset{w^{*}}{\to}0$, we get $\mu\circ\sigma^{-1}=\mu$ and then $\mu\in\mathcal{M}_{\sigma}(S_{\beta})$. It follows from

[TABLE]

that $\mu\in\mathcal{M}_{\beta,a}$, which contradicts $\mathcal{M}_{\beta,a}=\varnothing$.

(2) If $\mathcal{M}_{\beta,a}\neq\varnothing$, by Lemma 4.3, it suffices to prove that $\mathcal{M}_{\beta,a}$ is a closed connected set in $\mathcal{M}_{\sigma}(S_{\beta})$.

• \footnotesize{$1$}⃝

Prove that $\mathcal{M}_{\beta,a}$ is closed.

Let $\{\mu_{n},n\in\mathbb{N}\}\subset\mathcal{M}_{\beta,a}$ and $\mu\in\mathcal{M}_{\sigma}(S_{\beta})$ such that $\mu_{n}\overset{w^{*}}{\to}\mu$. It follows from

[TABLE]

that $\mu\in\mathcal{M}_{\beta,a}$.

• \footnotesize{$2$}⃝

Prove that $\mathcal{M}_{\beta,a}$ is connected.

It suffices to prove that $\mathcal{M}_{\beta,a}$ is path connected. In fact, for any $\mu_{0},\mu_{1}\in\mathcal{M}_{\beta,a}$, we define the path $f:[0,1]\to\mathcal{M}_{\beta,a}$ by $f(s):=\mu_{s}:=(1-s)\mu_{0}+s\mu_{1}$ for $s\in[0,1]$. Then $f(0)=\mu_{0}$, $f(1)=\mu_{1}$ and $f([0,1])\subset\mathcal{M}_{\beta,a}$. It remains to show that $f$ is continuous. Let $\{s,s_{n},n\geq 1\}\subset[0,1]$ such that $s_{n}\to s$. We only need to prove $f(s_{n})\to f(s)$, i.e., $\mu_{s_{n}}\overset{w^{*}}{\to}\mu_{s}$. Let $\varphi:S_{\beta}\to\mathbb{R}$ be a continuous function. It suffices to check $\int\varphi$ $d\mu_{s_{n}}\to\int\varphi$ $d\mu_{s}$, i.e.,

[TABLE]

This follows immediately from $s_{n}\to s$.

∎

Proof of Theorem 1.2.

By Proposition 4.1 it suffices to consider the following (1), (2) and (3).

(1) We have

[TABLE]

Since the first inequality is obvious, we only prove the second one as follows. Let $\mu\in\mathcal{M}_{\sigma}(S_{\beta})$ such that $\mu[0]=a$. Restricted to $\{[w]:w\in\cup_{n=1}^{m}\Sigma_{\beta}^{n}\}$, $\mu$ is obviously an $(\beta,m,a)$-coordinated set function. It suffices to prove $h_{\mu}(\sigma)\leq\mathfrak{h}_{\mu}(\beta,m)$. Using $\mathcal{P}:=\{[0],[1]\}$ as a partition generator of the Borel sigma-algebra on $(S_{\beta},d_{\beta})$, by simple calculation, we get that the conditional entropy of $\mathcal{P}$ given $\bigvee_{k=1}^{m-1}\sigma^{-k}\mathcal{P}$ with respect to $\mu$, denoted by H_{\mu}\Big{(}\mathcal{P}\mid\bigvee_{k=1}^{m-1}\sigma^{-k}\mathcal{P}\Big{)}, is equal to $\mathfrak{h}_{\mu}(\beta,m)$. Since H_{\mu}\Big{(}\mathcal{P}\mid\bigvee_{k=1}^{n-1}\sigma^{-k}\mathcal{P}\Big{)} decreases as $n$ increases and [31, Theorem 4.14] says that it converges to $h_{\mu}(\sigma)$, we get $h_{\mu}(\sigma)\leq\mathfrak{h}_{\mu}(\beta,m)$. In the following we attached the calculation mentioned above.

[TABLE]

(2) Prove

[TABLE]

[math] follows from the facts that every $(m-1)$-Markov measure $\mu\in M_{\sigma}(S_{\beta})$ with $\mu[0]=a$ restricted to $\{[w]:w\in\cup_{n=1}^{m}\Sigma_{\beta}^{n}\}$ is an $(\beta,m,a)$-coordinated set function and Proposition 2.11 implies $h_{\mu}(\sigma)=\mathfrak{h}_{\mu}(\beta,m)$.

[math] Let $\mu$ be an $(\beta,m,a)$-coordinated set function. By the entropy formula Proposition 2.11, it suffices to show that $\mu$ can be extended to be an $(m-1)$-Markov measure in $\mathcal{M}_{\sigma}(S_{\beta})$. Note that $\mu$ is already defined on all the cylinders of order $\leq m$. Suppose that for some $n\geq m$, $\mu$ is already defined on $\{[w_{1}\cdots w_{n}]:w_{1}\cdots w_{n}\in\Sigma_{\beta}^{n}\}$. Then we define

[TABLE]

where the right hand side is regarded as [math] if one of $\mu[w_{1}\cdots w_{n}]$, $\mu[w_{n-m+2}\cdots w_{n}]$ and $\mu[w_{n-m+2}\cdots w_{n+1}]$ is [math]. By Proposition 2.9 it suffices to check

[TABLE]

for all $w\in\Sigma_{\beta}^{n}$ with $n\geq m$. (Note that for $n\leq m-1$, \footnotesize{1}⃝ and \footnotesize{2}⃝ are already guaranteed by the condition that $\mu$ is $(\beta,m,a)$-coordinated.)

\footnotesize{1}⃝ Let $n\geq m$ and $w_{1}\cdots w_{n}\in\Sigma_{\beta}^{n}$.

• i)

If $w_{1}\cdots w_{n}1\in\Sigma_{\beta}^{*}$, then

[TABLE]

where ($\star$) can be proved as follows.

\footnotesize{a}⃝ If $\mu[w_{1}\cdots w_{n}]=0$, then ($\star$) is obvious.

\footnotesize{b}⃝ If $\mu[w_{n-m+2}\cdots w_{n}]=0$, since the fact that $\mu$ is $(\beta,m,a)$-coordinated implies $\mu[w_{n-m+1}\cdots w_{n}]\leq\mu[w_{n-m+2}\cdots w_{n}]$, we get $\mu[w_{n-m+1}\cdots w_{n}]=0$. Then

[TABLE]

and ($\star$) follows.

\footnotesize{c}⃝ If $\mu[w_{1}\cdots w_{n}]\neq 0$ and $\mu[w_{n-m+2}\cdots w_{n}]\neq 0$, then ($\star$) follows from

[TABLE]

noting that $\mu$ is $(\beta,m,a)$-coordinated.

• ii)

If $w_{1}\cdots w_{n}1\notin\Sigma_{\beta}^{*}$, by Proposition 2.3 and $w_{1}\cdots w_{n}\in\Sigma_{\beta}^{*}$ we get $w_{n-m+2}\cdots w_{n}1\notin\Sigma_{\beta}^{*}$. Since $\mu$ is $(\beta,m,a)$-coordinated, we get $\mu[w_{n-m+2}\cdots w_{n}0]=\mu[w_{n-m+2}\cdots w_{n}]$ and then

[TABLE]

where ($\star$) follows in the same way as i) \footnotesize{b}⃝ if $\mu[w_{n-m+2}\cdots w_{n}]=0$.

\footnotesize{2}⃝ Prove $\sum_{\begin{subarray}{c}u=0,1\\ uw_{1}\cdots w_{n}\in\Sigma_{\beta}^{*}\end{subarray}}\mu[uw_{1}\cdots w_{n}]=\mu[w_{1}\cdots w_{n}]$ for all $w_{1}\cdots w_{n}\in\Sigma_{\beta}^{n}$ and $n\geq m$ by induction. Since $\mu$ is $(\beta,m,a)$-coordinated, the conclusion is true for $n=m-1$. Now suppose that the conclusion is already true for some $n\geq m-1$. We consider $n+1$ in the following. Let $w_{1}\cdots w_{n+1}\in\Sigma_{\beta}^{n+1}$.

• i)

If $1w_{1}\cdots w_{n+1}\in\Sigma_{\beta}^{*}$, then $1w_{1}\cdots w_{n}\in\Sigma_{\beta}^{*}$ and

[TABLE]

where ($\star$) follows from inductive hypothesis.

• ii)

If $1w_{1}\cdots w_{n+1}\notin\Sigma_{\beta}^{*}$, by Proposition 2.3 and $w_{1}\cdots w_{n+1}\in\Sigma_{\beta}^{*}$ we get $1w_{1}\cdots w_{n}\notin\Sigma_{\beta}^{*}$, and then

[TABLE]

where ($\star$) follows from inductive hypothesis.

(3) By the definition of $(\beta,m,a)$-coordinated set functions and $\mathfrak{h}_{\mu}(\beta,m)$, it is straightforward to see that the supremum of

[TABLE]

can be achieved as a maximum. ∎

5. Proof of Theorem 1.3

We need the following lemma which follows immediately from the convexity of the function $x\log x$.

Lemma 5.1**.**

Let $\varphi:[0,\infty)\to\mathbb{R}$ be defined by

[TABLE]

Then for all $x,y\in[0,\infty)$ and $a,b\geq 0$ with $a+b=1$,

[TABLE]

The equality holds if and only if $x=y$, $a=0$ or $b=0$.

Proof of Theorem 1.3.

(1) By $\varepsilon^{*}(1,\beta)=(1^{m-1}0)^{\infty}$ and Lemma 2.2, we know that for any $x\in[0,1)$, every $m$ consecutive digits in $\varepsilon(x,\beta)$ must contain at least one [math]. This implies

[TABLE]

for all $n\in\mathbb{N}$, and then

[TABLE]

for any $x\in[0,1)$. If $0\leq a<\frac{1}{m}$, we get $F_{\beta,a}=\varnothing$.

(2) When $\frac{1}{m}\leq a\leq 1$, $f_{a}$ is a continuous function on its domain of definition

[TABLE]

which is closed and non-empty since

[TABLE]

Therefore $\max_{(x_{1},\cdots,x_{m-2})\in D_{m,a}}f_{a}(x_{1},\cdots,x_{m-2})$ exists.

In order to get our conclusion, by Theorem 1.2, it suffices to prove

[TABLE]

in the following \footnotesize{$1$}⃝ and \footnotesize{$2$}⃝, which are enlightened by drawing figures of the cylinders in $[0,1)$ and understanding their relations.

\footnotesize{$1$}⃝ Prove the inequality “$\leq$” in (5.1).

Let $\mu$ be an $(\beta,m,a)$-coordinated set function. By Lemma 2.2 we get $\Sigma_{\beta}^{m}=\{0,1\}^{m}\setminus\{1^{m}\}$, $\mu[1^{m-1}0]=\mu[1^{m-1}]$ and then

[TABLE]

For $i_{2}\cdots i_{m-1}\neq 1^{m-2}$ and $i_{m}\in\{0,1\}$, we can prove

[TABLE]

In fact, if $\mu[0i_{2}\cdots i_{m-1}]=0$, then $\mu[0i_{2}\cdots i_{m}]=0$. We get $\mu[1i_{2}\cdots i_{m-1}]=\mu[i_{2}\cdots i_{m-1}]-\mu[0i_{2}\cdots i_{m-1}]=\mu[i_{2}\cdots i_{m-1}]$ and $\mu[1i_{2}\cdots i_{m}]=\mu[i_{2}\cdots i_{m}]-\mu[0i_{2}\cdots i_{m}]=\mu[i_{2}\cdots i_{m}]$, which imply (5.2). If $\mu[1i_{2}\cdots i_{m-1}]=0$, in the same way we can get (5.2). If $\mu[0i_{2}\cdots i_{m-1}]\neq 0$ and $\mu[1i_{2}\cdots i_{m-1}]\neq 0$, then $\mu[i_{2}\cdots i_{m-1}]\neq 0$ and (5.2) follows from

[TABLE]

where the last inequality follows from Lemma 5.1. Thus

[TABLE]

For $i_{2}\cdots i_{m-2}\neq 1^{m-3}$ and $i_{m-1}\in\{0,1\}$, in the same way as proving (5.2), we get

[TABLE]

Thus

[TABLE]

$\cdots$

Repeat the above process a finite number of times. Finally we get

[TABLE]

Since $\mu$ is $(\beta,m,a)$-coordinated, we have

[TABLE]

Let $y_{1}:=\mu[01],y_{2}:=\mu[011],\cdots,y_{m-2}:=\mu[01^{m-2}]$. Then we have

[TABLE]

By a simple calculation, we get

[TABLE]

It follows from $\mu[00],\mu[010],\cdots,\mu[01^{m-3}0],\mu[01^{m-2}0],\mu[01^{m-1}]\geq 0$ that $(y_{1},\cdots,y_{m-2})\in D_{m,a}$. Therefore

[TABLE]

\footnotesize{$2$}⃝ Prove that the inequality “$\leq$” in (5.1) can achieve “$=$” by some $(\beta,m,a)$-coordinated set function.

Let $(y_{1},\cdots,y_{m-2})\in D_{m,a}$ such that

[TABLE]

Define

[TABLE]

[TABLE]

and

[TABLE]

where $\mu[uwv]$ is defined to be [math] if one of $\mu[w]$, $\mu[uw]$ and $\mu[wv]$ is [math]. Then $\mu$ is an $(\beta,m,a)$-coordinated set function. By (5.3) and Lemma 5.1, it is straightforward to check that in the proof of \footnotesize{$1$}⃝, all the “$\leq$” in the upper bound estimation of $\mathfrak{h}_{\mu}(\beta,m)$ can take “$=$” and then

[TABLE]

∎

Acknowledgement*.*

The author is grateful to Professor Jean-Paul Allouche and Professor Bing Li for their advices on a former version of this paper, and also grateful to the Oversea Study Program of Guangzhou Elite Project (GEP) for financial support (JY201815).