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Uniform approximation problems of expanding Markov maps

Published online by Cambridge University Press:  15 February 2023

YUBIN HE*
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, P. R. China
LINGMIN LIAO
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P. R. China (e-mail: [email protected])
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Abstract

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $, we investigate the size of the uniform approximation set

$$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$

The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $-almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $, where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $, we show that for $ \mu _\phi $-a.e. $ x $, the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $.

MSC classification

Type
Original Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press

1 Introduction and motivation

Denote by $ \|\cdot \| $ the distance to the nearest integer. The famous Dirichlet theorem asserts that for any real number $ x\in [0,1] $ and $ N\ge 1 $ , there exists a positive integer $ n $ such that

(1.1) $$ \begin{align} \|nx\|<\frac{1}{N}\quad\text{and}\quad n\le N. \end{align} $$

As a corollary, for any $ x\in [0,1] $ , there exist infinitely many positive integers $ n $ such that

(1.2) $$ \begin{align} \|nx\|\le\frac{1}{n}. \end{align} $$

Let $ (\{nx\})_{n\ge 0} $ be the orbit of $0$ of the rotation by an irrational number $ x $ , where $ \{nx\} $ is the fractional part of $ nx $ . From the dynamical perspective, the Dirichlet theorem and its corollary describe the rate at which $ 0 $ is approximated by the orbit $ (\{nx\})_{n\ge 0} $ in a uniform and asymptotic way, respectively.

In general, one can study the Hausdorff dimension of the set of points which are approximated by the orbit $ (\{nx\})_{n\ge 0} $ with a faster speed. For the asymptotic approximation, Bugeaud [Reference Bugeaud8] and, independently, Schmeling and Troubetzkoy [Reference Schmeling and Troubetzkoy28] proved that for any $ x\in [0,1]\setminus \mathbb Q $ ,

$$ \begin{align*}\operatorname{\mathrm{\dim_H}} \{y\in[0,1]:\|nx-y\|<n^{-\kappa}\text{ for infinitely many }n\}=\frac{1}{\kappa},\end{align*} $$

where $ \operatorname {\mathrm {\dim _H}} $ stands for the Hausdorff dimension. The corresponding uniform approximation problem was recently studied by Kim and Liao [Reference Kim and Liao18] who obtained the Hausdorff dimension of the set

$$ \begin{align*}\{y\in[0,1]:\text{ for all } N\gg1,\text{ there exists } n\le N, \text{ such that }\|nx-y\|<N^{-\kappa}\}.\end{align*} $$

Naturally, one wonders about the analog results when the orbit $ (\{nx\})_{n\ge 0} $ is replaced by an orbit $ (T^nx)_{n\ge 0} $ of a general dynamical system $ ([0,1], T) $ . For any $ \kappa>0 $ , Fan, Schmeling, and Troubetzkoy [Reference Fan, Schmeling and Troubetzkoy13] considered the set

$$ \begin{align*}\mathcal L^{\kappa}(x):=\{y\in[0,1]:|T^nx-y|<n^{-\kappa}\text{ for infinitely many }n\}\end{align*} $$

of points that are asymptotically approximated by the orbit $ (T^nx)_{n\ge 0} $ with a given speed $ n^{-\kappa } $ , where $ T $ is the doubling map. It seems difficult to investigate the size of $ \mathcal L^{\kappa }(x) $ when $ x $ is not a dyadic rational, as the distribution of $ (T^nx)_{n\ge 0} $ is not as well studied as that of $ (\{nx\})_{n\ge 0} $ , see for example [Reference Alessandri and Berthé1] for more details about the distribution of $ (\{nx\})_{n\ge 0} $ . However, from the viewpoint of ergodic theory, Fan, Schmeling, and Troubetzkoy [Reference Fan, Schmeling and Troubetzkoy13] obtained the Hausdorff dimension of $ \mathcal L^{\kappa }(x) $ for $ \mu _\phi $ almost all points $ x $ , where $ \mu _\phi $ is the Gibbs measure associated with a Hölder continuous potential $ \phi $ . They found that the size of $ \mathcal L^{\kappa }(x) $ is closely related to the local dimension of $ \mu _\phi $ and to the first hitting time for shrinking targets.

In their paper [Reference Liao and Seuret21], Liao and Seuret extended the results of [Reference Fan, Schmeling and Troubetzkoy13] to expanding Markov maps. Later, Persson and Rams [Reference Persson and Rams24] considered more general piecewise expanding interval maps, and proved some similar results to those of [Reference Fan, Schmeling and Troubetzkoy13, Reference Liao and Seuret21]. These studies are also closely related to the metric theory of a random covering set; see [Reference Barral and Fan3, Reference Dvoretzky11, Reference Fan12, Reference Jonasson and Steif17, Reference Seuret29, Reference Shepp30, Reference Tang32] and references therein.

As a counterpart of the dynamically defined asymptotic approximation set $ \mathcal L^{\kappa }(x) $ , we would like to study the corresponding uniform approximation set $ \mathcal U^\kappa (x) $ defined as

$$ \begin{align*}\begin{split} \mathcal U^\kappa(x):&=\{y\in[0,1]:\text{ for all } N\gg1,\text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}\\ &=\bigcup_{i=1}^\infty\bigcap_{N= i}^\infty\bigcup_{n=1}^N B(T^nx, N^{-\kappa}), \end{split}\end{align*} $$

where $ B(x,r) $ is the open ball with center $ x $ and radius $ r $ , and $ T $ is an expanding Markov map (see Definition 1.1).

As the studies on $ \mathcal L^\kappa (x) $ , we are interested in the sizes (Lebesgue measure and Hausdorff dimension) of $ \mathcal U^\kappa (x) $ . By a simple argument, one can check that $ \mathcal U^\kappa (x)\setminus \{T^nx\}_{n\ge 0}\subset \mathcal L^\kappa (x) $ . Thus trivially, one has $ \unicode{x3bb} (\mathcal U^\kappa (x)) \le \unicode{x3bb} (\mathcal L^\kappa (x))$ and $ \operatorname {\mathrm {\dim _H}} \mathcal U^\kappa (x)\le \operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x) $ . Here, $ \unicode{x3bb} $ denotes the Lebesgue measure on $ [0,1] $ .

Our first result asserts that for any $ \kappa>0 $ , the Lebesgue measure and the Hausdorff dimension of $ \mathcal U^\kappa (x) $ are constants almost surely with respect to a $ T $ -invariant ergodic measure.

Theorem 1.1. Let $ T $ be an expanding Markov map on $ [0,1] $ and $ \nu $ be a $ T $ -invariant ergodic measure. Then for any $ \kappa>0 $ , both $ \unicode{x3bb} (\mathcal U^\kappa (x)) $ and $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ are constants almost surely.

To further describe the size of $\mathcal U^\kappa (x)$ for almost all points, we impose a stronger condition, the same as that of Fan, Schmeling, and Troubetzkoy [Reference Fan, Schmeling and Troubetzkoy13] and Liao and Seuret [Reference Liao and Seuret21], that $ \nu $ is a Gibbs measure. Precisely, let $\phi $ be a Hölder continuous potential and $ \mu _\phi $ be its associated Gibbs measure. Let $ \mathcal B^\kappa (x):=[0,1]\setminus \mathcal U^\kappa (x) $ . We remark that the sets $\mathcal U^\kappa (x)$ are decreasing (the sets $\mathcal B^\kappa (x)$ are increasing) with respect to $\kappa $ . Then, we want to ask, for $\mu _\phi $ almost all points x, how the size of $\mathcal U^\kappa (x)$ (and $\mathcal B^\kappa (x)$ ) changes with respect to $\kappa $ . We thus would like to answer the following questions.

  1. (Q1) When $\mathcal U^\kappa (x)=[0,1]$ for $\mu _\phi $ -almost every (a.e.) x, what is the critical value

    $$ \begin{align*} \kappa_{\phi}:=\sup\{\kappa\geq 0: \mathcal U^\kappa(x)=[0,1] \ \text{for }\mu_\phi\text{-a.e.} \,x\}? \end{align*} $$
  2. (Q2) When $\unicode{x3bb} (\mathcal U^\kappa (x))=1$ for $\mu _\phi $ -a.e. x, what is the critical value

    $$ \begin{align*} \kappa_{\phi}^\unicode{x3bb}:=\sup\{\kappa \geq 0: \unicode{x3bb}(\mathcal U^\kappa(x))=1 \ \text{for }\mu_\phi\text{-a.e.} \,x\}? \end{align*} $$
  3. (Q3) What are the Hausdorff dimensions of $\mathcal U^\kappa (x)$ and $\mathcal B^\kappa (x)$ for $ \mu _\phi $ -a.e. $ x $ ? What is the critical value

    $$ \begin{align*} \kappa_{\phi}^H:=\sup\{\kappa \geq 0: \operatorname{\mathrm{\dim_H}}(\mathcal U^\kappa(x))=0 \ \text{for }\mu_\phi\text{-a.e.} \,x\}? \end{align*} $$

In this paper, we answer these questions when $ T $ is an expanding Markov map of the interval $ [0,1] $ with a finite partition—an expanding Markov map, for short. We stress that according to our definition, an expanding Markov map is mixing (see the following Theorem 4.2 whose proof can be found in [Reference Baladi2, Reference Liverani, Saussol and Vaienti22, Reference Parry and Pollicott23, Reference Ruelle27]).

Definition 1.1. (Expanding Markov map)

A transformation $ T:[0,1]\to [0,1] $ is an expanding Markov map with a finite partition provided that there is a partition of $ [0,1] $ into subintervals $ I(i)=(a_i, a_{i+1}) $ for $ i=0,\ldots ,Q-1 $ with endpoints $ 0=a_0<a_1<\cdots <a_Q=1 $ satisfying the following properties.

  1. (1) There is an integer $ n_0 $ and a real number $ \rho $ such that $ |(T^{n_0})'|\ge \rho>1 $ .

  2. (2) $ T $ is strictly monotonic and can be extended to a $ C^2 $ function on each $ \overline {I(i)} $ .

  3. (3) If $ I(j)\cap T(I(k))\ne \emptyset $ , then $ I(j)\subset T(I(k)) $ .

  4. (4) There is an integer $ R $ such that $ I(j)\subset \bigcup _{n=1}^R T^n(I(k)) $ for every $ k $ , $ j $ .

  5. (5) For every $ k\in \{0,1,\ldots ,Q-1\} $ , $ \sup _{(x,y,z)\in I(k)^3}(|T"(x)|/|T'(y)||T'(z)|)<\infty $ .

For a probability measure $ \nu $ and for $ y\in [0,1] $ , we set

$$ \begin{align*}\underline d_\nu(y):=\liminf_{r\to 0}\frac{\log\nu(B(y,r))}{\log r}\quad\text{and}\quad\overline d_\nu(y):=\limsup_{r\to 0}\frac{\log\nu(B(y,r))}{\log r},\end{align*} $$

which are called respectively the lower and upper local dimensions of $ \nu $ at $ y $ . When $ \underline d_\nu (y)=\overline d_\nu (y) $ , their common value is denoted by $ d_\nu (y) $ , and is simply called the local dimension of $ \nu $ at $ y $ . Let $ D_\nu $ be the multifractal spectrum of $ \nu $ defined by

$$ \begin{align*}D_\nu(s):= \operatorname{\mathrm{\dim_H}}\{y\in[0,1]:d_\nu(y)=s\} \quad \text{for all } s\in\mathbb{R}.\end{align*} $$

Our answers to questions (Q1)–(Q3) are stated in the following theorem. For an expanding Markov map $ T $ and a Hölder continuous potential $ \phi $ , we first define

(1.3) $$ \begin{align} \alpha_-:&=\min_{\nu\in\mathcal M_{\mathrm{inv}}}\frac{-\int\phi \,d\nu}{\int\log|T'|\,d\nu}, \end{align} $$
(1.4) $$ \begin{align} \alpha_{\max}:&=\frac{-\int\phi \, d\mu_{\max}}{\int\log|T'|\, d\mu_{\max}}, \end{align} $$
(1.5) $$ \begin{align} \alpha_+:&=\max_{\nu\in\mathcal M_{\mathrm{inv}}}\frac{-\int\phi \, d\nu}{\int\log|T'|\, d\nu}, \end{align} $$

where $ \mathcal M_{\mathrm {inv}} $ is the set of $ T $ -invariant probability measures on $ [0,1] $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $ . By definition, it holds that $ \alpha _-\le \alpha _{\max }\le \alpha _+ $ . Indeed, the quantities $ \alpha _- $ , $ \alpha _{\max } $ , and $ \alpha _+ $ depend on $ T $ and $ \phi $ . However, for simplicity, we leave out the dependence unless the context requires specification.

The following main theorem tells us that the three critical values demanded in questions (Q1)–(Q3) are $ 1/\alpha _+ $ , $ 1/\alpha _{\max } $ , and $ 1/\alpha _-$ , correspondingly.

Theorem 1.2. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure.

  1. (1) The critical value $ \kappa _{\phi } $ is $ 1/\alpha _+ $ . Namely, for $ \mu _\phi $ -a.e. $ x $ , $ \mathcal U^\kappa (x)=[0,1] $ if $ 1/\kappa>\alpha _+ $ , and $ \mathcal U^\kappa (x)\ne [0,1] $ if $ 1/\kappa <\alpha _+ $ .

  2. (2) The critical value $ \kappa _{\phi }^\unicode{x3bb} $ is $ 1/\alpha _{\max } $ . Moreover, for $ \mu _\phi $ -a.e. $ x $ ,

    $$ \begin{align*}\unicode{x3bb}(\mathcal U^\kappa(x))=1-\unicode{x3bb}(\mathcal B^\kappa(x))=\begin{cases} 0\quad&\text{if }1/\kappa\in(0,\alpha_{\max}),\\ 1\quad&\text{if }1/\kappa\in(\alpha_{\max},+\infty). \end{cases}\end{align*} $$
  3. (3) The critical value $ \kappa _{\phi }^H $ is $ 1/\alpha _{-} $ . Moreover, for $ \mu _\phi $ -a.e. $ x $ ,

    $$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mathcal U^\kappa(x)=\begin{cases} D_{\mu_\phi}(1/\kappa) &\text{if }1/\kappa\in(0,\alpha_{\max}]\setminus\{\alpha_-\},\\ 1&\text{if }1/\kappa\in(\alpha_{\max},+\infty). \end{cases}\end{align*} $$
  4. (4) For $ \mu _\phi $ -a.e. $ x $ ,

    $$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mathcal B^\kappa(x)=\begin{cases} 1&\text{if }1/\kappa\in(0,\alpha_{\max}),\\ D_{\mu_\phi}(1/\kappa) &\text{if }1/\kappa\in[\alpha_{\max},+\infty)\setminus \{\alpha_+\}. \end{cases}\end{align*} $$

Remark 1. Item (2) of Theorem 1.2 is valid in a wider setting (in particular, the Markov assumption can be dropped) and is a direct consequence of [Reference Holland, Kirsebom, Kunde and Persson15, Theorem 3.2] and [Reference Kirsebom, Kunde and Persson19, Proposition 1 and Theorem 5]. We will provide a self-contained proof for the reader’s convenience.

Remark 2. It is worth noting that the multifractal spectrum $ D_{\mu _\phi }(s) $ vanishes if $ s\notin [\alpha _-,\alpha _+] $ . So if $ 1/\kappa <\alpha _- $ , then $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=0 $ for $ \mu _\phi $ -a.e. $ x $ .

Remark 3. The cases $ 1/\kappa =\alpha _- $ and $ \alpha _+ $ are not covered by Theorem 1.2. However, if the multifractal spectrum $ D_{\mu _\phi } $ is continuous at $ \alpha _- $ (respectively $ \alpha _+ $ ), we get that $ \operatorname {\mathrm {\dim _H}}\mathcal U^{\alpha _-}(x)=0 $ (respectively $ \operatorname {\mathrm {\dim _H}}\mathcal B^{\alpha _+}(x)=0 $ ) for $ \mu _\phi $ -a.e. $ x $ . The situation becomes more subtle if $ D_{\mu _\phi }(\cdot ) $ is discontinuous at $ \alpha _- $ (respectively $ \alpha _+ $ ). Our methods do not work for obtaining the value of $ \operatorname {\mathrm {\dim _H}}\mathcal U^{\alpha _-}(x) $ (respectively $ \operatorname {\mathrm {\dim _H}}\mathcal B^{\alpha _+}(x)=0 $ ) for $ \mu _\phi $ -a.e. $ x $ .

Let $ \operatorname {\mathrm {\dim _H}} \nu $ be the dimension of the Borel probability measure $ \nu $ defined by

$$ \begin{align*}\operatorname{\mathrm{\dim_H}} \nu=\inf\{\operatorname{\mathrm{\dim_H}} E:E\text{ is a Borel set of }[0,1]\text{ and }\nu(E)>0\}.\end{align*} $$

Remark 4. As already discussed above, $ \mathcal U^\kappa (x)\setminus \{T^nx\}_{n\ge 0}\subset \mathcal L^\kappa (x) $ , one may wonder whether the sets $ \mathcal U^\kappa (x) $ and $ \mathcal L^\kappa (x) $ are essentially different. More precisely, is it possible that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ is strictly less than $ \operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x) $ ? Theorem 1.2 affirmatively answers this question. Compared with the asymptotic approximation set $ \mathcal L^{\kappa }(x) $ , the structure of the uniform approximation set $ \mathcal U^\kappa (x) $ does have a notable feature. When $ 1/\kappa \in (0,\operatorname {\mathrm {\dim _H}}\mu _\phi )\setminus \{\alpha _-\} $ , the map $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ agrees with the multifractal spectrum $ D_{\mu _\phi }(1/\kappa ) $ , while the map $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal L^{\kappa }(x) $ is the linear function $ f(1/\kappa )=1/\kappa $ independent of the multifractal spectrum. Therefore, $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)<\operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x) $ . See Figure 1 for an illustration.

Figure 1 The multifractal spectrum of $ \mu _\phi $ and the maps $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal B^\kappa (x) $ , $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ , and $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x) $ .

Remark 5. For the asymptotic approximation set $ \mathcal L^\kappa (x) $ , the most difficult part lies in establishing the lower bound for $ \operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x)$ when $ 1/\kappa <\operatorname {\mathrm {\dim _H}}\mu _\phi $ , for which a multifractal mass transference principle for Gibbs measure is applied, see [Reference Fan, Schmeling and Troubetzkoy13, §8], [Reference Liao and Seuret21, §5.2], and [Reference Persson and Rams24, §6]. Specifically, since $ \mu _\phi (\mathcal L^{\delta }(x))=1 $ for all $ 1/\delta>\operatorname {\mathrm {\dim _H}}\mu _\phi $ , the multifractal mass transference principle guarantees the lower bound $ \operatorname {\mathrm {\dim _H}} \mathcal L^{\kappa }(x)\ge (\operatorname {\mathrm {\dim _H}}\mu _\phi )\delta /\kappa $ for all $ 1/\kappa <\operatorname {\mathrm {\dim _H}}\mu _\phi $ . By letting $ 1/\delta $ monotonically decrease to $ \operatorname {\mathrm {\dim _H}}\mu _\phi $ along a sequence $ (\delta _n) $ , we get immediately the expected lower bound $ \operatorname {\mathrm {\dim _H}} \mathcal L^{\kappa }(x)\ge 1/\kappa $ for all $ 1/\kappa <\operatorname {\mathrm {\dim _H}}\mu _\phi $ . However, recent progresses in uniform approximation [Reference Bugeaud and Liao9, Reference Kim and Liao18, Reference Koivusalo, Liao and Persson20, Reference Zheng and Wu34] indicate that there is no mass transference principle for uniform approximation set. Therefore, we cannot expect that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ decreases linearly with respect to $ 1/\kappa $ as $ \operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x) $ does. The main new ingredient of this paper is the difficult upper bound for $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ when $ 1/\kappa <\operatorname {\mathrm {\dim _H}}\mu _\phi $ . To overcome the difficulty, we fully develop and combine the methods in [Reference Fan, Schmeling and Troubetzkoy13, Reference Koivusalo, Liao and Persson20].

To illustrate our main theorem, let us give some examples.

Example 1. Suppose that $ T $ is the doubling map and $ \mu _\phi :=\unicode{x3bb} $ is the Lebesgue measure. Applying Theorem 1.2, we have that for $ \unicode{x3bb} $ -a.e. $ x $ ,

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mathcal U^\kappa(x)=\begin{cases} 0 &\text{if }1/\kappa\in(0,1),\\ 1&\text{if }1/\kappa\in(1,+\infty). \end{cases}\end{align*} $$

The Lebesgue measure is monofractal and hence the corresponding multifractal spectrum $ D_\unicode{x3bb} $ is discontinuous at $ 1 $ . Theorem 1.2 fails to provide any metric statement for the set $ \mathcal U^1(x) $ for $ \unicode{x3bb} $ -a.e. $ x $ . However, we can conclude that $ \mathcal U^1(x) $ is a Lebesgue null set for $ \unicode{x3bb} $ -a.e. $ x $ from the Fubini theorem and a zero-one law established in [Reference Holland, Kirsebom, Kunde and Persson15, Theorem 2.1]. Further, by Theorem 1.1, $ \operatorname {\mathrm {\dim _H}}\mathcal U^1(x) $ is Lebesgue almost surely a constant.

Some sets similar to $ \mathcal U^1(x) $ in Example 1 have recently been studied by Koivusalo, Liao, and Persson [Reference Koivusalo, Liao and Persson20]. In their paper, instead of the orbit $ (T^nx)_{n\ge 0} $ , they investigated the sets of points uniformly approximated by an independent and identically distributed sequence $ (\omega _n)_{n\ge 1} $ . Specifically, they showed that with probability one, the lower bound of the Hausdorff dimension of the set

$$ \begin{align*}\begin{split} \{y\in[0,1]:\text{ for all } N\gg1,\text{ there exists } n\le N, \text{ such that }|\omega_n-y|<1/N\} \end{split}\end{align*} $$

is larger than $ 0.217\,744\,429\,848\,5995 $ [Reference Koivusalo, Liao and Persson20, Theorem 5].

Example 2. Let $ p\in (1/2,1) $ . Suppose that $ T $ is the doubling map on $ [0,1] $ and $ \mu _p $ is the $ (p,1-p) $ Bernoulli measure. It is known that the multifractal spectrum $ D_{\mu _p} $ is continuous on $ (0,+\infty ) $ and attains its unique maximal value $ 1 $ at $ -\log _2 (p(1-p))/2 $ . Theorem 1.2 then gives that for $ \mu _p $ -a.e. $ x $ ,

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mathcal U^\kappa(x)=\begin{cases} D_{\mu_p}(1/\kappa) &\text{if }1/\kappa\in\bigg(0,\frac{-\log_2 (p(1-p))}{2}\bigg),\\[8pt] 1&\text{if }1/\kappa\in\bigg[\frac{-\log_2 (p(1-p))}{2},+\infty\bigg). \end{cases}\end{align*} $$

Our paper is organized as follows. We start in §2 with some preparations on an expanding Markov map, and then use ergodic theory to give a proof of Theorem 1.1. Section 3 contains some recollections on multifractal analysis and a variational principle which are essential in the proof of Theorem 1.2. Section 4 describes some relations among hitting time, approximating rate, and local dimension of $ \mu _\phi $ . From these relations, we then derive items (1), (2), and (4) of Theorem 1.2 in §5.1, as well as the lower bound of $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ in §5.2. In the same §5.2, we establish the upper bound of $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ , which is arguably the most substantial part.

2 Basic definitions and the proof of Theorem 1.1

2.1 Covering of $ [0,1] $ by basic intervals

Let $ T $ be an expanding Markov map as defined in Definition 1.1. For each $ (i_1i_2\cdots i_n)\in \{0,1,\ldots ,Q-1\}^n $ , we call

$$ \begin{align*}I(i_1i_2\cdots i_n):=I(i_1)\cap T^{-1}(I(i_2))\cap\cdots\cap T^{-n+1}(I(i_n))\end{align*} $$

a basic interval of generation $ n $ . It is possible that $ I(i_1i_2\cdots i_n) $ is empty for some $ (i_1i_2\cdots i_n)\in \{0,1,\ldots ,Q-1\}^n $ . The collection of non-empty basic intervals of a given generation $ n $ will be denoted by $ \Sigma _n $ . Let $ \mathcal E $ denote the set of endpoints of basic intervals. The set $ \mathcal E $ is a countable set, so $ \operatorname {\mathrm {\dim _H}} \mathcal E=0 $ . For any $ x\in [0,1]\setminus \mathcal E $ , we denote $ I_n(x) $ the unique basic interval $ I\in \Sigma _n $ containing $ x $ .

By the definition of an expanding Markov map, we obtain the following bounded distortion property on basic intervals: there is a constant $ L>1 $ such that for any $ x\in [0,1]\setminus \mathcal E $ ,

(2.1) $$ \begin{align} \text{for any }n\ge 1,\quad L^{-1}|(T^n)'(x)|^{-1}\le |I_n(x)|\le L|(T^n)'(x)|^{-1}, \end{align} $$

where $ |I| $ is the length of the interval $ I $ . Consequently, we can find two constants $ 1<L_1<L_2 $ such that

(2.2) $$ \begin{align} \text{for every } I\in \Sigma_n,\quad L_2^{-n}\le |I|\le L_1^{-n}. \end{align} $$

2.2 Proof of Theorem 1.1

Let us start with a simple but crucial observation.

Lemma 2.1. Let $ T $ be an expanding Markov map. For any $ x\in [0,1] $ and $ \kappa>0 $ , we have $ \mathcal U^\kappa (x)\setminus \{Tx\}\subset \mathcal U^{\kappa }(Tx) $ .

Proof. Let $ y\in \mathcal U^\kappa (x)\setminus \{Tx\} $ and let $ i $ be the smallest integer satisfying $ y\notin B(Tx, i^{-\kappa }) $ . By the definition of $ \mathcal U^\kappa (x) $ , for any integer $ N> i $ large enough, there exists $ 1\le n\le N $ such that $ y\in B(T^nx, N^{-\kappa }) $ . Moreover, the condition $ N> i $ implies that $ n\ne 1 $ , and hence $ y\in B(T^{n-1}(Tx),N^{-\kappa }) $ . This gives that $ y\in \mathcal U^\kappa (Tx) $ ; therefore, $ \mathcal U^\kappa (x)\setminus \{Tx\}\subset \mathcal U^{\kappa }(Tx) $ .

Recall that a $ T $ -invariant measure $ \nu $ is ergordic if and only if any $ T $ -invariant function is constant almost surely. The proof of Theorem 1.1 falls naturally into two parts. We first deal with the Lebesgue measure part.

Proof of Theorem 1.1: Lebesgue measure part

For any $ \kappa>0 $ , define the function $ g_\kappa :x\mapsto \unicode{x3bb} (\mathcal U^\kappa (x)) $ . We claim that $ g_\kappa $ is measurable. In fact, it suffices to observe that

$$ \begin{align*}g_\kappa(x)=\lim_{i\to \infty}\lim_{m\to\infty}\unicode{x3bb}\bigg(\bigcap_{N=i}^m\bigcup_{n=1}^NB(T^nx,N^{-\kappa})\bigg)\end{align*} $$

and that, by the piecewise continuity of $ T^n $ ( $ n\ge 1 $ ), the set

$$ \begin{align*}\bigg\{x\in[0,1]:\unicode{x3bb}\bigg(\bigcap_{N=i}^m\bigcup_{n=1}^NB(T^nx,N^{-\kappa})\bigg)>t\bigg\}\end{align*} $$

is measurable for any $ t\in \mathbb R $ .

By Lemma 2.1, we see that $ \unicode{x3bb} (\mathcal U^\kappa (Tx))\ge \unicode{x3bb} (\mathcal U^\kappa (x)) $ , or equivalently, $ g_\kappa (Tx)\ge g_\kappa (x) $ . Since $ g_\kappa $ is measurable, by the fact $ 0\le g_\kappa \le 1 $ and the invariance of $ \nu $ , we have that $ g_\kappa $ is invariant with respect to $ \nu $ , that is, $ g_\kappa (Tx)=g_\kappa (x) $ for $ \nu $ -a.e. $ x $ . In the presence of ergodicity of $ \nu $ , $ g_\kappa $ is constant almost surely.

Proof of Theorem 1.1: Hausdorff dimension part

Fix $ \kappa>0 $ and define the function $ f_\kappa :x\mapsto \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ . Again by Lemma 2.1, we see that $ f_\kappa (Tx)\ge f_\kappa (x) $ . Proceeding in the same way as the Lebesgue measure part, we get that $ f_\kappa $ is constant almost surely provided that $ f_\kappa $ is measurable.

To show that $ f_\kappa $ is measurable, it suffices to prove that for any $ t>0 $ , the set

$$ \begin{align*}\begin{split} A(t):=\{x\in[0,1]:f_\kappa(x)<t\}=\bigg\{x\in[0,1]:\operatorname{\mathrm{\dim_H}}\!\bigg(\bigcup_{i=1}^\infty\bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\!\bigg)\!<t\bigg\} \end{split}\end{align*} $$

is measurable. Throughout the proof of this part, we will assume that the ball $ B(T^nx, N^{-\kappa }) $ is closed. This makes the proof achievable and it does not change the Hausdorff dimension of $ \mathcal U^\kappa (x) $ .

By the definition of Hausdorff dimension, a point $ x\in A(t) $ if and only if there exists $ h\in \mathbb N $ such that

$$ \begin{align*}\mathcal H^{t-{1}/{h}}\bigg(\bigcup_{i=1}^\infty\bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)=0,\end{align*} $$

or equivalently for all $ i\ge 1 $ ,

(2.3) $$ \begin{align} \mathcal H^{t-{1}/{h}}\bigg(\bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)=0. \end{align} $$

By the definition of Hausdorff measure, equation (2.3) holds if and only if for any $ j,k\in \mathbb N $ ,

$$ \begin{align*}\mathcal H_{{1}/{j}}^{t-{1}/{h}}\bigg(\bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)<\frac{1}{k}.\end{align*} $$

Hence, we see that

$$ \begin{align*} A(t)&=\bigcup_{h=1}^\infty\bigcap_{i=1}^\infty\bigcap_{j=1}^\infty\bigcap_{k=1}^\infty\bigg\{ x\in [0,1]:\mathcal H_{{1}/{j}}^{t-{1}/{h}}\bigg(\bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)<\frac{1}{k}\bigg\}\\&=:\bigcup_{h=1}^\infty\bigcap_{i=1}^\infty\bigcap_{j=1}^\infty\bigcap_{k=1}^\infty B_{h,i,j,k}. \end{align*} $$

If $ x\in B_{h,i,j,k} $ , then there is a countable open cover $ \{U_p\}_{p\ge 1} $ with $ 0<|U_p|<1/j $ satisfying

(2.4) $$ \begin{align} \bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\subset \bigcup_{p=1}^\infty U_p\quad \text{and}\quad\sum_{p=1}^\infty |U_p|^{t-{1}/{h}}<\frac{1}{k}. \end{align} $$

The set $ \bigcap _{N= i}^\infty \bigcup _{n=1}^NB(T^nx, N^{-\kappa }) $ can be viewed as the intersection of a family of decreasing compact sets $ \{\bigcap _{N= i}^l\bigcup _{n=1}^NB(T^nx, N^{-\kappa })\}_{l\ge i} $ , and hence there exists $ l_0\ge i $ satisfying

$$ \begin{align*}\bigcap_{N= i}^{l_0}\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\subset\bigcup_{p=1}^\infty U_p,\end{align*} $$

which implies

$$ \begin{align*}\mathcal H_{{1}/{j}}^{t-{1}/{h}}\bigg(\bigcap_{N= i}^{l_0}\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)<\frac1k.\end{align*} $$

We then deduce that

(2.5) $$ \begin{align} B_{h,i,j,k}\subset\bigcup_{l=i}^{\infty}\bigg\{ x\in [0,1]:\mathcal H_{{1}/{j}}^{t-{1}/{h}}\bigg(\bigcap_{N= i}^l\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)<\frac1k\bigg\}=:\bigcup_{l=i}^{\infty}C_{h,i,j,k,l}. \end{align} $$

If $ x\in C_{h,i,j,k,l} $ for some $ l\ge 1 $ , then

$$ \begin{align*}\mathcal H_{{1}/{j}}^{t-{1}/{h}}\bigg(\bigcap_{N= i}^\infty\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)\le\mathcal H_{{1}/{j}}^{t-{1}/{h}}\bigg(\bigcap_{N= i}^l\bigcup_{n=1}^NB(T^nx, N^{-\kappa})\bigg)<\frac{1}{k}.\end{align*} $$

Hence, $ x\in B_{i,j,k,l} $ and the reverse inclusion of equation (2.5) is proved.

Notice that $ T,T^2,\ldots ,T^l $ are continuous on every basic interval of generation greater than $ l $ . For any $ x\in C_{h,i,j,k,l} $ , denote $ \mathcal S(x):=\bigcap _{N= i}^l\bigcup _{n=1}^NB(T^nx, N^{-\kappa }) $ . There is an open cover $ (V_p)_{p\ge 1} $ of $ \mathcal S(x) $ with $ 0<|V_p|<1/j $ and $ \sum _p |V_p|^{t-1/h}<1/k $ . Since $ \mathcal S(x) $ is compact, we see that the distance $ \delta $ between $ \mathcal S(x) $ and the complement of $ \bigcup _{p\ge 1} V_p $ is positive. By the continuities of $ T,T^2,\ldots ,T^l $ , there is some $ l_1:=l_1(\delta )>l $ for which if $ y\in I_{l_1}(x) $ , then $ \mathcal S(y) $ is contained in the $ \delta /2 $ -neighborhood of $ \mathcal S(x) $ . Thus, $ \mathcal S(y) $ can also be covered by $ \bigcup _{p\ge 1} V_p $ and $ I_{l_1}(x)\subset C_{h,i,j,k,l} $ . Finally, $ C_{h,i,j,k,l} $ is a union of some basic intervals, which is measurable.

Now, combining the equalities obtained above, we have

$$ \begin{align*}A(t)=\bigcup_{h=1}^\infty\bigcap_{i=1}^\infty\bigcap_{j=1}^\infty\bigcap_{k=1}^\infty\bigcup_{l=i}^{\infty}C_{h,i,j,k,l},\end{align*} $$

which is a Borel measurable set.

3 Multifractal properties of Gibbs measures

In this section, we review some of the standard facts on multifractal properties of Gibbs measures.

Definition 3.1. A Gibbs measure $ \mu _\phi $ associated with a potential $ \phi $ is a probability measure satisfying the following: there exists a constant $ \gamma>0 $ such that

$$ \begin{align*}\text{for any basic interval }I\in\Sigma_n, \quad\gamma^{-1}\le\frac{\mu_\phi(I)}{e^{S_n\phi(x)-nP(\phi)}}\le\gamma\quad\text{for every }x\in I,\end{align*} $$

where $ S_n\phi (x)=\phi (x)+\cdots +\phi (T^{n-1}x) $ is the $ n $ th Birkhoff sum of $ \phi $ at $ x $ , and $ P(\phi ) $ is the topological pressure of $\phi $ defined by

$$ \begin{align*}P(\phi)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{I\in\Sigma_n}\sup_{x\in I}e^{S_n\phi(x)}.\end{align*} $$

The following theorem ensures the existence and uniqueness of invariant Gibbs measure.

Theorem 3.1. [Reference Bowen6, Reference Walters33]

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map. Then for any Hölder continuous function $ \phi $ , there exists a unique $ T $ -invariant Gibbs measure $ \mu _\phi $ associated with $ \phi $ . Further, $ \mu _\phi $ is ergodic.

The Gibbs measure $ \mu _\phi $ also satisfies the quasi-Bernoulli property (see [Reference Liao and Seuret21, Lemma 4.1]), that is, for any $ n>k\ge 1 $ , for any basic interval $ I(i_1\cdots i_n)\in \Sigma _{n} $ , we have

(3.1) $$ \begin{align} \gamma^{-3}\mu_\phi(I')\mu_\phi(I")\le\mu_\phi(I)=\mu_\phi(I'\cap T^{-k}I")\le \gamma^3\mu_\phi(I')\mu_\phi(I"), \end{align} $$

where $ I'=I(i_1\cdots i_k)\in \Sigma _k $ and $ I"=I(i_{k+1}\cdots i_n)\in \Sigma _{n-k} $ . It follows immediately that

(3.2) $$ \begin{align} \text{for any }m\ge k,\quad\mu_\phi(I'\cap T^{-m}U)\le \gamma^3\mu_\phi(I')\mu_\phi(U), \end{align} $$

where $ U $ is an open set in $ [0,1] $ .

We adopt the convention that $ \phi $ is normalized, that is, $ P(\phi )=0 $ . If it is not the case, we can replace $\phi $ by $ \phi -P(\phi ) $ .

Now, let us recall some standard facts on multifractal analysis which aims at studying the multifractal spectrum $ D_{\mu _\phi } $ . Some multifractal analysis results were summarized in [Reference Liao and Seuret21] and we present them as follows. The proofs can also be found in Refs. [Reference Barreira, Pesin and Schmeling4, Reference Brown, Michon and Peyrière7, Reference Collet, Lebowitz and Porzio10, Reference Pesin and Weiss25, Reference Rand26, Reference Simpelaere31].

Theorem 3.2. [Reference Liao and Seuret21, Theorem 2.5]

Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure. Then, the following hold.

  1. (1) The function $ D_{\mu _\phi } $ of $ \mu _\phi $ is a concave real-analytic map on the interval $ (\alpha _-,\alpha _+) $ , where $ \alpha _- $ and $ \alpha _+ $ are defined in equations (1.3) and (1.5), respectively.

  2. (2) The spectrum $ D_{\mu _\phi } $ reaches its maximum value $ 1 $ at $ \alpha _{\max } $ defined in equation (1.4).

  3. (3) The graph of $ D_{\mu _\phi } $ and the line with equation $ y=x $ intersect at a unique point which is $ (\operatorname {\mathrm {\dim _H}}\mu _\phi ,\operatorname {\mathrm {\dim _H}}\mu _\phi ) $ . Moreover, $ \operatorname {\mathrm {\dim _H}}\mu _\phi $ satisfies

    $$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mu_\phi=\frac{-\int\phi\, d\mu_\phi}{\int\log|T'|\, d\mu_\phi}.\end{align*} $$

Proposition 3.3. [Reference Liao and Seuret21, §2.3]

For every $ q\in \mathbb R $ , there is a unique real number $ \eta _\phi (q) $ such that the topological pressure $ P(-\eta _\phi (q)\log |T'|+q\phi ) $ is equal to $ 0 $ . Further, $ \eta _\phi (q) $ is real-analytic and concave.

Remark 6. For simplicity, denote by $ \mu _q $ the $ T $ -invariant Gibbs measure associated with the potential $ -\eta _\phi (q)\log |T'|+q\phi $ . Certainly, $ \eta _\phi (0)=1 $ and the corresponding measure $ \mu _0 $ is associated with the potential $ -\log |T'| $ . By the bounded distortion property in equation (2.1), the Gibbs measure $ \mu _0 $ , coinciding with $ \mu _{\max } $ , is strongly equivalent to the Lebesgue measure $\unicode{x3bb} $ , which means that there exists a constant $ c\ge 1 $ such that for every measurable set $ E $ , we have

$$ \begin{align*}{c^{-1}\mu_0(E)\le\unicode{x3bb}(E)\le c\mu_0(E).}\end{align*} $$

For every $ q\in \mathbb R $ , we introduce the exponent

(3.3) $$ \begin{align} \alpha(q)=\frac{-\int\phi\, d\mu_q}{\int\log|T'|\, d\mu_q}. \end{align} $$

Proposition 3.4. [Reference Liao and Seuret21, §2.3]

Let $ \mu _q $ and $ \alpha (q) $ be as above. The following statements hold.

  1. (1) The Gibbs measure $ \mu _q $ is supported by the level set $ \{y:d_{\mu _\phi }(y)=\alpha (q)\} $ and $ D_{\mu _\phi }(\alpha (q))=\operatorname {\mathrm {\dim _H}}\mu _q=\eta _\phi (q)+q\alpha (q) $ .

  2. (2) The map $ \alpha (q) $ is decreasing, and

    $$ \begin{align*} \lim_{q\to+\infty}\alpha(q)=\alpha_-,&\quad\lim_{q\to-\infty}\alpha(q)=\alpha_+,\notag\\ \alpha(1)=\operatorname{\mathrm{\dim_H}}\mu_\phi,&\ \ \quad\alpha(0)=\alpha_{\max}. \end{align*} $$
  3. (3) The inverse of $ \alpha (q) $ exists and is denoted by $ q(\alpha ) $ . Moreover, $ q(\alpha )<0 $ if $ \alpha \in (\alpha _{\max },\alpha _+) $ , and $ q(\alpha )\ge 0 $ if $ \alpha \in (\alpha _-,\alpha _{\max }] $ .

Recall that $ \mathcal E $ is the set of endpoints of basic intervals which has Hausdorff dimension 0. For a measure $ \nu $ and a point $ y\in [0,1]\setminus \mathcal E $ , define the lower and upper Markov pointwise dimensions respectively by

$$ \begin{align*}\underline{M}_{\nu}(y):=\liminf_{n\to\infty}\frac{\log\nu(I_n(y))}{\log |I_n(y)|},\quad \overline{M}_{\nu}(y):=\limsup_{n\to\infty}\frac{\log\nu(I_n(y))}{\log |I_n(y)|}.\end{align*} $$

When $ \underline {M}_{\nu }(y)=\overline {M}_{\nu }(y) $ , their common value is denoted by $ M_\nu (y) $ . By equations (2.1) and (2.2), we have

$$ \begin{align*}\overline d_{\nu}(y)\le \overline{M}_{\nu}(y),\end{align*} $$

which implies the inclusions

(3.4) $$ \begin{align} {\{y:d_{\nu}(y)=s\}\setminus \mathcal{E} \subset\{y:\underline d_{\nu}(y)\ge s\}\setminus \mathcal{E}\subset\{y:\overline d_{\nu}(y)\ge s\}\setminus \mathcal{E}\subset\{y:\overline{M}_{\nu}(y)\ge s\}.} \end{align} $$

By the Gibbs property of $ \mu _\phi $ and the bounded distortion property on basic intervals in equation (2.1), the definitions of Markov pointwise dimensions can be reformulated as

(3.5) $$ \begin{align} \overline{M}_{\mu_\phi}(y)=\limsup_{n\to \infty}\frac{S_n\phi(y)}{S_n(-\log T')(y)}\quad\text{and}\quad M_{\mu_\phi}(y)=\lim_{n\to \infty}\frac{S_n\phi(y)}{S_n(-\log T')(y)}. \end{align} $$

This allows us to derive the following lemma, which is an alternative version of a proposition due to Jenkinson [Reference Jenkinson16, Proposition 2.1]. We omit its proof since the argument is similar.

Lemma 3.5. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure. Then,

$$ \begin{align*}\sup_{y\in [0,1]}\overline{M}_{\mu_\phi}(y)=\sup_{y\colon M_{\mu_\phi}(y)\text{ exists }}M_{\mu_\phi}(y)=\max_{\nu\in \mathcal M_{\mathrm{inv}} }\frac{-\int\phi \, d\nu}{\int\log|T'|\, d\nu}=\alpha_+.\end{align*} $$

In particular, for any $ s>\alpha _+ $ ,

$$ \begin{align*}\{y:d_{\mu_\phi}(y)=s\}=\{y:\overline d_{\mu_\phi}(y)\ge s\}=\emptyset.\end{align*} $$

We finish the section with a variational principle.

Lemma 3.6. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure.

  1. (1) For every $ s<\alpha _{\max } $ , $ \operatorname {\mathrm {\dim _H}}\{y:\underline d_{\mu _\phi }(y)\le s\}=\operatorname {\mathrm {\dim _H}}\{y:\overline d_{\mu _\phi }(y)\le s\}=D_{\mu _\phi }(s) $ .

  2. (2) For every $ s\in (\alpha _{\max },+\infty )\setminus \alpha _+ $ , $ \operatorname {\mathrm {\dim _H}}\{y:\underline d_{\mu _\phi }(y)\ge s\}=\operatorname {\mathrm {\dim _H}}\{y:\overline d_{\mu _\phi }(y)\ge s\}=D_{\mu _\phi }(s) $ .

Proof. (1) Note that

$$ \begin{align*} \{y:d_{\mu_\phi}(y)=s\}\subset\{y:\overline d_{\mu_\phi}(y)\le s\}\subset\{y:\underline d_{\mu_\phi}(y)\le s\}. \end{align*} $$

In [Reference Liao and Seuret21, Proposition 2.8], the leftmost set and the rightmost set were shown to have the same Hausdorff dimension. This together with the above inclusions completes the proof of the first point of the lemma.

(2) When $ T $ is the doubling map, the statement was formulated by Fan, Schmeling, and Trobetzkoy [Reference Fan, Schmeling and Troubetzkoy13, Theorem 3.3]. Our proof follows their idea closely, we include it for completeness.

By Lemma 3.5, we can assume without loss of generality that $ s<\alpha _+ $ . The inclusions in equation (3.4) imply the following inequalities:

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\{y:d_{\mu_\phi}(y)=s\} \le \operatorname{\mathrm{\dim_H}}\{y:\overline d_{\mu_\phi}(y)\ge s\}\le\operatorname{\mathrm{\dim_H}}\{y:\overline{M}_{\mu_\phi}(y)\ge s\}.\end{align*} $$

We turn to prove the reverse inequalites. By Proposition 3.4 and the condition $ s>\alpha _{\max } $ , there exists a real number $ q_s:=q(s)<0 $ such that

$$ \begin{align*}s=\frac{-\int\phi\, d\mu_{q_s}}{\int\log|T'|\, d\mu_{q_s}} \quad\text{ and }\quad D_{\mu_\phi}(s)=\operatorname{\mathrm{\dim_H}}\mu_{q_s}=\eta_\phi(q_s)+q_ss,\end{align*} $$

where $ \mu _{q_s} $ is the Gibbs measure associated with the potential $ -\eta _\phi (q_s)\log |T'|+q_s\phi $ . Now let $ y $ be any point such that $ \overline {M}_{\mu _\phi }(y)\ge s $ . By Proposition 3.3, the topological pressure $ P(-\eta _\phi (q_s)\log |T'|+q_s\phi ) $ is $ 0 $ . Then, we can apply the Gibbs property of $ \mu _{q_s} $ and equation (2.1) to yield

$$ \begin{align*}\begin{split} \underline{M}_{\mu_{q_s}}(y)&=\liminf_{n\to\infty}\frac{\log e^{S_n(-\eta_\phi(q_s)\log|T'|+q_s\phi)(y)}}{\log|I_n(y)|}\\ &=\liminf_{n\to\infty}\bigg(\frac{-\eta_\phi(q_s)\log|(T^n)'(y)|}{\log|I_n(y)|}+q_s\cdot\frac{\log e^{S_n\phi(y)}}{\log|I_n(y)|}\bigg)\\ &=\eta_\phi(q_s)+q_s\cdot\limsup_{n\to\infty}\frac{\log\mu_\phi (I_n(y))}{\log|I_n(y)|}\\ &=\eta_\phi(q_s)+q_s \overline{M}_{\mu_\phi}(y)\\ &\le \eta_\phi(q_s)+q_ss=D_{\mu_\phi}(s), \end{split}\end{align*} $$

where the inequality holds because $ q_s<0 $ .

Finally, Billingsley’s lemma [Reference Bishop and Peres5, Lemma 1.4.1] gives

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\{y:\overline{M}_{\mu_\phi}(y)&\ge s\}\le\operatorname{\mathrm{\dim_H}}\{y:\underline{M}_{\mu_{q_s}}(y)\le D_{\mu_\phi}(s)\}\le D_{\mu_\phi}(s)\\&=\operatorname{\mathrm{\dim_H}}\{y:d_{\mu_\phi}(y)=s\}.\\[-3.1pc] \end{align*} $$

4 Covering questions related to hitting time and local dimension

In §4.1, we reformulate the uniform approximation set $ \mathcal U^\kappa (x) $ in terms of hitting time. Thereafter, we relate the first hitting time for shrinking balls to local dimension in §4.2.

4.1 Covering questions and hitting time

Denote $ \mathcal O^+(x):=\{T^nx:n\ge 1\} $ .

Definition 4.1. For every $ x,y\in [0,1] $ and $ r>0 $ , define the first hitting time of the orbit of $ x $ into the ball $ B(y,r) $ by

$$ \begin{align*}\tau_r(x,y):=\inf\{n\ge 1:T^nx\in B(y,r)\}.\end{align*} $$

Set

$$ \begin{align*}\underline{R}(x,y):=\liminf_{r\to 0}\frac{\log\tau_r(x,y)}{-\log r}\quad\text{and}\quad\overline{R}(x,y):=\limsup_{r\to 0}\frac{\log\tau_r(x,y)}{-\log r}.\end{align*} $$

For convenience, when $ \mathcal O^+(x)\cap B(y,r)=\emptyset $ , we set $ \tau _r(x,y)=\infty $ and $ \underline {R}(x,y)=\overline {R}(x,y) =\infty $ . If $ \underline {R}(x,y)=\overline {R}(x,y) $ , we denote the common value by $ R(x,y) $ .

For any ball $ B\subset [0,1] $ , define the first hitting time $ \tau (x, B) $ by

$$ \begin{align*}\tau(x, B):=\inf \{n\ge 1:T^nx\in B\}.\end{align*} $$

Similarly, we set $ \tau (x, B)=\infty $ when $ \mathcal O^+(x)\cap B=\emptyset $ .

The following lemma exhibits a relation between $ \mathcal U^\kappa (x) $ and hitting time.

Lemma 4.1. For any $ \kappa>0 $ , we have

$$ \begin{align*} \bigg\{ y\in[0,1]:\overline{R}(x,y)>\frac{1}{\kappa} \bigg\}&\subset\mathcal B^\kappa(x)\subset\bigg\{ y\in[0,1]:\overline{R}(x,y)\ge\frac{1}{\kappa} \bigg\},\\ \bigg\{ y\in[0,1]:\overline{R}(x,y)<\frac{1}{\kappa} \bigg\}&\subset\mathcal U^\kappa(x)\subset\bigg\{ y\in[0,1]:\overline{R}(x,y)\le\frac{1}{\kappa} \bigg\}. \end{align*} $$

Proof. The top left and bottom right inclusions imply one another. Let us prove the bottom right inclusion. Suppose that $ y\in \mathcal U^\kappa (x) $ . Then for all large enough $ N $ , there is an $ n\le N $ such that $ T^nx \in B(y,N^{-\kappa })$ . Thus, $ \tau _{N^{-\kappa }}(x,y)\le N $ for all $ N $ large enough, which implies $ \overline {R}(x,y)\le 1/\kappa $ .

The top right and bottom left inclusions imply one another. So, it remains to prove the bottom left inclusion. Consider $ y $ such that $ \overline {R}(x,y)<1/\kappa $ . If $ y\in \mathcal O^+(x) $ with $ y=T^{n_0}x $ for some $ n_0\ge 1 $ , then the system

$$ \begin{align*}|T^nx-y|=|T^nx-T^{n_0}x|<N^{-\kappa}\quad\text{and}\quad 1\le n\le N\end{align*} $$

always has a trivial solution $ n=n_0 $ for all $ N\ge n_0 $ . Therefore, $ y\in \mathcal U^\kappa (x) $ . Now assume that $ y\notin \mathcal O^+(x) $ . By the definition of $ \overline {R}(x,y) $ , there is a positive real number $ r_0<1 $ such that

$$ \begin{align*}\tau_r(x,y)<r^{-1/\kappa} \quad\text{for all }0<r<r_0.\end{align*} $$

Denote $ n_r:=\tau _r(x,y) $ for all $ 0<r<r_0 $ . Since $ y\notin \mathcal O^+(x) $ , the family of positive integers $ \{n_r:0<r<r_0\} $ is unbounded. For each $ N> r_0^{-1/\kappa } $ , denote $ t:=N^{-\kappa } $ . The definition of $ n_{t} $ implies that

$$ \begin{align*}T^{n_{t}}x\in B(y,t)=B(y,N^{-\kappa}).\end{align*} $$

We conclude $ y\in \mathcal U^\kappa (x) $ by noting that $ n_{t}<t^{-1/\kappa }=N $ .

4.2 Relation between hitting time and local dimension

As Lemma 4.1 shows, we need to study the hitting time $ \overline {R}(x,y) $ of the Gibbs measure $ \mu _\phi $ . We will prove that the hitting time is related to local dimension when the measure is exponential mixing.

Definition 4.2. A $ T $ -invariant measure $ \nu $ is exponential mixing if there exist two constants $ C>0 $ and $ 0<\beta <1 $ such that for any ball $ A $ and any Borel measurable set $ B $ ,

(4.1) $$ \begin{align} |\nu(A\cap T^{-n}B)-\nu(A)\nu(B)|\le C\beta^n\nu(B). \end{align} $$

Theorem 4.2. [Reference Baladi2, Reference Liverani, Saussol and Vaienti22, Reference Parry and Pollicott23, Reference Ruelle27]

The $ T $ -invariant Gibbs measure $ \mu _\phi $ associated with a Hölder continuous potential $ \phi $ of an expanding Markov map $ T $ is exponential mixing.

The exponential mixing property allows us to apply the following theorem which describes a relation between hitting time and local dimension of invariant measure.

Theorem 4.3. [Reference Galatolo14]

Let $ (X,T,\nu ) $ be a measure-theoretic dynamical system. If $ \nu $ is superpolynomial mixing and if $ d_\nu (y) $ exists, then for $ \nu $ -a.e. $ x $ , we have

$$ \begin{align*}R(x,y)=d_\nu(y).\end{align*} $$

It should be noticed that the superpolynomial mixing property is much weaker than the exponential mixing property.

Now, we turn to the study of the expanding Markov map $ T $ on the interval $ [0,1] $ . An application of Fubini’s theorem yields the following corollary.

Corollary 4.4. [Reference Liao and Seuret21, Corollary 3.8]

Let $ T $ be an expanding Markov map. Let $ \mu _\phi $ and $ \mu _\psi $ be two $ T $ -invariant Gibbs probability measures on $ [0,1] $ associated with Hölder potentials $ \phi $ and $ \psi $ , respectively. Then,

$$ \begin{align*}\text{for }\mu_\phi\times\mu_\psi\text{-a.e.\,}(x,y),\quad R(x,y)=d_{\mu_\phi}(y)=\frac{-\int\phi\, d\mu_\psi}{\int\log|T'|\, d\mu_\psi}.\end{align*} $$

5 The studies of $ \mathcal B^\kappa (x) $ and $ \mathcal U^\kappa (x) $

5.1 The study of $ \mathcal B^\kappa (x) $

In this subsection, we are going to prove Theorem 1.2 except for item (3). Let us start with the lower bound for $ \operatorname {\mathrm {\dim _H}}\mathcal B^\kappa (x) $ .

Lemma 5.1. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure. For any $ \kappa>0 $ , the following hold.

  1. (a) If $ 1/\kappa \in (0,\alpha _{\max }) $ , then $ \unicode{x3bb} (\mathcal B^\kappa (x))=1 $ for $ \mu _\phi $ -a.e. $ x $ .

  2. (b) If $ 1/\kappa \in [\alpha _{\max },+\infty )\setminus \{\alpha _+\} $ , then $ \operatorname {\mathrm {\dim _H}}\mathcal B^\kappa (x)\ge D_{\mu _\phi }(1/\kappa ) $ for $ \mu _\phi $ -a.e. $ x $ .

Proof. (a) Let $ 1/\kappa \in (0,\alpha _{\max }) $ . As already observed in §3, the Gibbs measure $ \mu _0 $ associated with $ \log |T'| $ is strongly equivalent to the Lebesgue measure $ \unicode{x3bb} $ . Thus, a set $ F $ has full $ \mu _0 $ -measure if and only if $ F $ has full $ \unicode{x3bb} $ -measure. Corollary 4.4 implies that

$$ \begin{align*}\text{for }\mu_\phi\times\mu_0\text{-}\text{a.e.}\,(x,y),\quad R(x,y)=d_{\mu_\phi}(y)=\frac{-\int\phi\, d\mu_0}{\int\log|T'|\, d\mu_0}=\alpha_{\max}.\end{align*} $$

By Fubini’s theorem, for $ \mu _\phi $ -a.e. $ x $ , the set $ \{y:R(x,y)=d_{\mu _\phi }(y)=\alpha _{\max }\} $ has full $ \mu _0 $ -measure. Then for $ \mu _\phi $ -a.e. $ x $ , we have

$$ \begin{align*}\mu_0(\{ y:\overline{R}(x,y)>1/\kappa \})\ge \mu_0(\{y:R(x,y)=d_{\mu_\phi}(y)=\alpha_{\max}\})=1.\end{align*} $$

By Lemma 4.1, we arrive at the conclusion.

(b) By Lemma 3.5, the level set $ \{y:d_{\mu _\phi }(y)=1/\kappa \} $ is empty if $ 1/\kappa>\alpha _+ $ . Thus, $ D_{\mu _\phi }(1/\kappa )=0 $ and therefore $ \operatorname {\mathrm {\dim _H}}\mathcal B^\kappa (x)\ge 0=D_{\mu _\phi }(1/\kappa ) $ trivially holds for all $ 1/\kappa>\alpha _+ $ .

Let $ 1/\kappa \in [\alpha _{\max },\alpha _+) $ . We can suppose that $ \alpha _{\max }\ne \alpha _+ $ , since otherwise $ [\alpha _{\max },\alpha _+)=\emptyset $ and there is nothing to prove. For any $ s\in (1/\kappa ,\alpha _+) $ , by Proposition 3.4, there exists a real number $ q_s:=q(s) $ such that

$$ \begin{align*}s=\frac{-\int\phi\, d\mu_{q_s}}{\int\log|T'|\, d\mu_{q_s}}\quad\text{and}\quad\mu_{q_s}(\{y:d_{\mu_\phi}(y)=s\})=1.\end{align*} $$

Applying Corollary 4.4, we obtain

$$ \begin{align*}\text{for }\mu_\phi\times\mu_{q_s}\text{-}\text{a.e.}\,(x,y),\quad R(x,y)=d_{\mu_\phi}(y)=\frac{-\int\phi\, d\mu_{q_s}}{\int\log|T'|\, d\mu_{q_s}}=s.\end{align*} $$

It follows from Fubini’s theorem that, for $ \mu _\phi $ -a.e. $ x $ , the set $ \{y:R(x,y)=d_{\mu _\phi }(y)=s\} $ has full $ \mu _{q_s} $ -measure. Consequently, for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*}\begin{split} \operatorname{\mathrm{\dim_H}}\{y:\overline{R}(x,y)>1/\kappa\}&\ge \operatorname{\mathrm{\dim_H}}\{y:R(x,y)=d_{\mu_\phi}(y)=s\}\\ &\ge \operatorname{\mathrm{\dim_H}} \mu_{q_s}=D_{\mu_\phi}(s). \end{split}\end{align*} $$

We conclude by noting that $ s\in (1/\kappa ,\alpha _+) $ is arbitrary and $ D_{\mu _\phi } $ is continuous on $ [\alpha _{\max },\alpha _+) $ .

We are left to determine the upper bound of $ \operatorname {\mathrm {\dim _H}}\mathcal B^\kappa (x) $ . The following four lemmas were initially proved by Fan, Schmeling, and Troubetzkoy [Reference Fan, Schmeling and Troubetzkoy13] for the doubling map, and later by Liao and Sereut [Reference Liao and Seuret21] in the context of expanding Markov maps. We follow their ideas and demonstrate more general results. In Lemmas 5.25.4, we will not assume that $ T $ is an expanding Markov map.

Lemma 5.2. Let $ T $ be a map on $ [0,1] $ and $ \nu $ be a $ T $ -invariant exponential mixing measure. Let $ A_1,A_2,\ldots ,A_k $ be $ k $ subsets of $ [0,1] $ such that each $ A_i $ is a union of at most $ m $ disjoint balls. Then,

$$ \begin{align*}\prod_{i=1}^{k}\bigg(1-\frac{mC\beta^d}{\nu(A_i)}\bigg)\le \frac{\nu(A_1\cap T^{-d}A_2\cap\cdots\cap T^{-d(k-1)}A_k)}{\nu(A_1)\nu(A_2)\cdots\nu(A_k)}\le \prod_{i=1}^{k}\bigg(1+\frac{mC\beta^d}{\nu(A_i)}\bigg),\end{align*} $$

where $ \beta $ is the constant appearing in equation (4.1).

Proof. Since each $ A_i $ is a union of at most $ m $ disjoint balls, the exponential mixing property of $ \nu $ gives that, for every $ d\ge 1 $ ,

(5.1) $$ \begin{align} |\nu(A_i\cap T^{-d}B)-\nu(A_i)\nu(B)|\le mC\beta^d\nu(B), \end{align} $$

where $ B $ is a Borel measurable set. In particular, defining

$$ \begin{align*}B_i=A_i\cap T^{-d}A_{i+1}\cap\cdots\cap T^{-d(k-i)}A_k,\end{align*} $$

we get, for any $ i<k $ ,

$$ \begin{align*}|\nu(A_i\cap T^{-d}(B_{i+1}))-\nu(A_i)\nu(B_{i+1})|\le mC\beta^d\nu(B_{i+1}).\end{align*} $$

The above inequality can be written as

$$ \begin{align*} 1-\frac{mC\beta^d}{\nu(A_i)}\le \frac{\nu(A_i\cap T^{-d}B_{i+1})}{\nu(A_i)\nu(B_{i+1})}\le 1+\frac{mC\beta^d}{\nu(A_i)}. \end{align*} $$

Multiplying over all $ i\le k $ and using the identity

$$ \begin{align*}B_{i+1}=A_{i+1}\cap T^{-d}B_{i+2},\end{align*} $$

we have

$$ \begin{align*}\kern-18pt\prod_{i=1}^{k}\bigg(1-\frac{mC\beta^d}{\nu(A_i)}\bigg)\le \frac{\nu(A_1\cap T^{-d}A_2\cap\cdots\cap T^{-d(k-1)}A_k)}{\nu(A_1)\nu(A_2)\cdots\nu(A_k)}\le \prod_{i=1}^{k}\bigg(1+\frac{mC\beta^d}{\nu(A_i)}\bigg). \end{align*} $$

The following lemma illustrates that balls with small local dimension for exponential mixing measure are hit with big probability.

Lemma 5.3. Let $ T $ be a map on $ [0,1] $ and $ \nu $ be a $ T $ -invariant exponential mixing measure. Let $ h $ and $ \epsilon $ be two positive real numbers. For each $ n\in \mathbb N $ , consider $ N\le 2^n $ distinct balls $ B_1,\ldots ,B_N $ satisfying $ |B_i|=2^{-n} $ and $ \nu (B_i)\ge 2^{-n(h-\epsilon )} $ for all $ 1\le i\le N $ . Set

$$ \begin{align*}\mathcal C_{n,N,h}=\{x\in[0,1]:\text{ there exists } 1\le i\le N\text{ such that }\tau(x,B_i)\ge 2^{nh}\}.\end{align*} $$

Then there exists an integer $ n_h\in \mathbb N $ independent of $ N $ such that

$$ \begin{align*}\text{for every }n\ge n_h,\quad \nu(\mathcal C_{n,N,h})\le 2^{-n}.\end{align*} $$

Proof. For each $ i\le N $ , let

$$ \begin{align*}\Delta_{i}:=\{x\in[0,1]:\quad\text{ for all } k\le 2^{nh}, T^kx\notin B_i \}.\end{align*} $$

Obviously, we have $ \mathcal C_{n,N,h}=\bigcup _{i=1}^N\Delta _{i} $ , so it suffices to bound from above each $ \nu (\Delta _{i}) $ . Pick an integer $ \omega $ such that $ \omega>\log _{\beta ^{-1}}2^{h} $ . Let $ k=[2^{nh}/(\omega n)] $ be the integer part of $ 2^{nh}/(\omega n) $ . Then,

$$ \begin{align*}\Delta_{i}\subset\bigcap_{j=1}^{k}\{x\in[0,1]: T^{j\omega n}x\notin B_i\}=\bigcap_{j=1}^{k}T^{-j\omega n}B_i^c.\end{align*} $$

Since $ \omega>\log _{\beta ^{-1}}2^{h}$ , there is an $ n_h $ large enough such that for any $ n\ge n_h $ ,

(5.2) $$ \begin{align} 2C\beta^{\omega n}<2^{-nh-1}\le \nu(B_i)/2 \end{align} $$

and

(5.3) $$ \begin{align} 2^{n+1} \exp\bigg(\frac{-2^{n\epsilon}}{2\omega n}\bigg)\le 2^{-n}. \end{align} $$

Now applying Lemma 5.2 to $ A_l= B_i $ for all $ l\le N $ and to $ m=2 $ , we conclude from equation (5.2) that

$$ \begin{align*} \nu(\Delta_{i})\le\nu(\bigcap_{j=1}^{k}T^{-j\omega n}B_i^c) &\le (\nu(B_i^c)+2C\beta^{\omega n})^{k}\\ &\le (1-\nu(B_i)/2)^{k}\\ &\le (1-2^{-n(h-\epsilon)-1})^{ 2^{nh}/(\omega n)-1}\\ &=(1-2^{-n(h-\epsilon)-1})^{-1} \exp\bigg(\frac{2^{nh}\log(1-2^{-n(h-\epsilon)-1})}{\omega n}\bigg)\\ &\le 2\exp\bigg(\frac{-2^{n\epsilon}}{2\omega n}\bigg). \end{align*} $$

By equation (5.3),

$$ \begin{align*}\nu(C_{n,N,h})\le \sum_{i=1}^{N}\nu(B_i)\le 2^{n+1} \exp\bigg(\frac{-2^{n\epsilon}}{2\omega n}\bigg)\le 2^{-n}.\end{align*} $$

Let us recall that $ \{y:\overline {R}(x,y)\ge s\} $ is a random set depending on the random element $ x $ , but $ \{y:\overline d_{\mu _\phi }(y)\ge s\} $ is independent of $ x $ . The following lemma reveals a connection between the random set $ \{y:\overline {R}(x,y)\ge s\} $ and the deterministic set $ \{y:\overline d_{\mu _\phi }(y)\ge s\} $ .

Lemma 5.4. Let $ T $ be a map on $ [0,1] $ and $ \nu $ be a $ T $ -invariant exponential mixing measure. Let $ s\ge 0 $ . Then for $ \nu $ -a.e. $ x $ ,

$$ \begin{align*}\{y:\overline{R}(x,y)\ge s\}\subset\{y:\overline d_{\nu}(y)\ge s\}.\end{align*} $$

Proof. The case $ s=0 $ is obvious. We therefore assume $ s>0 $ . For any integer $ n\ge 1 $ , let

$$ \begin{align*}\mathcal R_{n,s,\epsilon}(x)&=\{y:\tau(x,B(y,2^{-n+1}))\ge 2^{n(s-\epsilon)}\},\\\mathcal E_{n,s,\epsilon}&=\{y:\nu(B(y,2^{-n}))\le 2^{-n(s-2\epsilon)}\}.\end{align*} $$

By definition, $ y\in \{y:\overline {R}(x,y)\ge s\} $ if and only if for any $ \epsilon>0 $ , there exist infinitely many integers $ n $ such that

$$ \begin{align*}\frac{\log \tau(x,B(y,2^{-n+1}))}{\log 2^n}\ge s-\epsilon.\end{align*} $$

Hence, we have

(5.4) $$ \begin{align} \{y:\overline{R}(x,y)\ge s\}=\bigcap_{\epsilon>0}\limsup_{n\to\infty}\mathcal R_{n,s,\epsilon}(x). \end{align} $$

Similarly,

$$ \begin{align*}\{y:\overline d_{\mu_\phi}(y)\ge s\}=\bigcap_{\epsilon>0}\limsup_{n\to\infty}\mathcal E_{n,s,\epsilon}.\end{align*} $$

Thus, it is sufficient to prove that, for $ \nu $ -a.e. $ x $ , there exists some integer $ n(x) $ such that

(5.5) $$ \begin{align} \text{for all }n\ge n(x),\quad\mathcal R_{n,s,\epsilon}(x)\subset \mathcal E_{n,s,\epsilon}, \end{align} $$

or equivalently,

(5.6) $$ \begin{align} \text{for all }n\ge n(x),\quad\mathcal E^c_{n,s,\epsilon}(x)\subset \mathcal R^c_{n,s,\epsilon}. \end{align} $$

Notice that $ \mathcal E_{n,s,\epsilon }^c $ can be covered by $ N\le 2^n $ balls with center in $ \mathcal E_{n,s,\epsilon }^c $ and radius $ 2^{-n} $ . Let $ \mathcal F_{n,s,\epsilon }:=\{B_1,B_2,\ldots ,B_N\} $ be the collection of these balls. By definition, we have $ \nu (B_i)\le 2^{-n(s-2\epsilon )} $ . Applying Lemma 5.3 to the collection $ \mathcal F_{n,s,\epsilon } $ of balls and to $ h=s-\epsilon $ , we see that

$$ \begin{align*}\sum_{n\ge n_h}\nu(\{x:\text{ there exists } B\in\mathcal F_{n,s,\epsilon} \text{ such that }\tau(x,B)\ge 2^{n(s-\epsilon)}\})\le \sum_{n\ge n_h}2^{-n}<\infty.\end{align*} $$

By the Borel–Cantelli lemma, for $ \nu $ -a.e. $ x $ , there exists an integer $ n(x) $ such that

$$ \begin{align*}\text{for all }n\ge n(x),\text{ for all }B\in \mathcal F_{n,s,\epsilon},\quad \tau(x, B)< 2^{n(s-\epsilon)}.\end{align*} $$

If $ y\in B $ for some $ B\in \mathcal F_{n,s,\epsilon } $ and $ n\ge n(x) $ , then $ B\subset B(y,2^{-n+1}) $ , which implies that $ \tau (x, B(y,2^{-n+1}))<\tau (x,B) $ . We then deduce that $ B $ is included in $ \mathcal R_{n,s,\epsilon }^c $ . This yields $ \mathcal E_{n,s,\epsilon }^c\subset \mathcal R_{n,s,\epsilon }^c $ , which is what we want.

Remark 7. With the notation in Lemma 5.4, proceeding with the same argument as equation (5.4), we have

$$ \begin{align*} \{y:\underline{R}(x,y)\ge s\}=\bigcap_{\epsilon>0}\liminf_{n\to\infty}\mathcal R_{n,s,\epsilon}(x)\quad \text{and}\quad\{y:\underline d_{\mu_\phi}(y)\ge s\}=\bigcap_{\epsilon>0}\liminf_{n\to\infty}\mathcal E_{n,s,\epsilon}.\end{align*} $$

It then follows from equation (5.5) that for $ \nu $ -a.e. $ x $ ,

$$ \begin{align*}\{y:\underline{R}(x,y)\ge s\}\subset\{y:\underline d_{\nu}(y)\ge s\}.\end{align*} $$

Applying Lemma 5.4 to the Gibbs measure $ \mu _\phi $ , we get the following upper bound.

Lemma 5.5. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure. Suppose $ 1/\kappa \ge \alpha _{\max } $ . Then for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mathcal B^\kappa(x)\le D_{\mu_\phi}(1/\kappa).\end{align*} $$

Moreover, if $ 1/\kappa>\alpha _+ $ , then for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*}\mathcal B^\kappa(x)=\emptyset.\end{align*} $$

Proof. Recall that Lemma 4.1 asserts that

$$ \begin{align*}\mathcal B^\kappa(x)\subset\{y:\overline{R}(x,y)\ge 1/\kappa\}\cup\mathcal O^+(x).\end{align*} $$

A direct application of Proposition 3.6 and Lemma 5.4 yields the first conclusion.

The second conclusion follows from Lemmas 3.5 and 4.1.

Collecting the results obtained in this subsection, we can prove Theorem 1.2 except for item (3).

Proof of the items (1), (2), and (4) of Theorem 1.2

Combining with Lemmas 5.1 and 5.5, we get the desired result.

5.2 The study of $ \mathcal U^\kappa (x) $

In this subsection, we prove the remaining part of Theorem 1.2, that is, item (3). We begin by showing the lower bound of $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ , which may be proved in much the same way as Lemma 5.1.

Lemma 5.6. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure.

  1. (a) If $ 1/\kappa \in (0,\alpha _{\max }]\setminus \{\alpha _-\} $ , then $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)\ge D_{\mu _\phi }(1/\kappa ) $ for $ \mu _\phi $ -a.e. $ x $ .

  2. (b) If $ 1/\kappa \in (\alpha _{\max },+\infty ) $ , then $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $ -a.e. $ x $ .

Proof. (a) By Lemma 3.6, the Hausdorff dimension of the level set $ \{y:d_{\mu _\phi }(y)=1/\kappa \} $ is zero if $ 1/\kappa <\alpha _- $ . Therefore, $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)\ge 0=D_{\mu _\phi }(1/\kappa ) $ .

The remaining case $ 1/\kappa \in (\alpha _-,\alpha _{\max }] $ holds by the same reasoning as proving Lemma 5.1(b).

(b) Observe that the full Lebesgue measure statement implies the full Hausdorff dimension statement. It follows from item (2) of Theorem 1.2 that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)= 1 $ when $ 1/\kappa \in (\alpha _{\max },+\infty ) $ .

It is left to show the upper bound of $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $ when $ 1/\kappa \le \alpha _{\max } $ . The proof combines the methods developed in [Reference Fan, Schmeling and Troubetzkoy13, §7] and [Reference Koivusalo, Liao and Persson20, Theorem 8]. Heuristically, the larger the local dimension of a point is, the less likely it is to be hit.

Lemma 5.7. Let $ T $ be an expanding Markov map. Let $ \phi $ be a Hölder continuous potential and $ \mu _\phi $ be the corresponding Gibbs measure. Let $ 1/\kappa \le \alpha _{\max } $ , then for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\mathcal U^\kappa(x)\le D_{\mu_\phi}(1/\kappa).\end{align*} $$

Proof. The proof will be divided into two steps.

Step 1. Given any $ a>1/\kappa $ , we are going to prove that

(5.7) $$ \begin{align} \operatorname{\mathrm{\dim_H}} (\mathcal U^\kappa(x)\cap\{y:\underline d_{\mu_\phi}(y)>a\})=0\quad\text{for }\mu_\phi\text{-}\text{a.e.}\,x. \end{align} $$

Suppose now that equation (5.7) is established. Let $ (a_m)_{m\ge 1} $ be a monotonically decreasing sequence of real numbers converging to $ 1/\kappa $ . Applying equation (5.7) to each $ a_m $ yields a full $ \mu _\phi $ -measure set corresponding to $ a_m $ . Then, by taking the intersection of these countable full $ \mu _\phi $ -measure sets, we conclude from the countable stability of Hausdorff dimension that

$$ \begin{align*}\operatorname{\mathrm{\dim_H}} (\mathcal U^\kappa(x)\cap\{y:\underline d_{\mu_\phi}(y)>1/\kappa\})=0\quad\text{for }\mu_\phi\text{-}\text{a.e.}\,x.\end{align*} $$

As a result, by Lemma 3.6, for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*} \operatorname{\mathrm{\dim_H}} \mathcal U^\kappa(x)&=\operatorname{\mathrm{\dim_H}} (\mathcal U^\kappa(x)\cap(\{y:\underline{d}_{\mu_\phi}(y)\le 1/\kappa\}\cup \{y:\underline d_{\mu_\phi}(y)>1/\kappa\}))\notag\\ &=\operatorname{\mathrm{\dim_H}} (\mathcal U^\kappa(x)\cap\{y:\underline{d}_{\mu_\phi}(y)\le 1/\kappa\})\notag\\ &\le\operatorname{\mathrm{\dim_H}}\{y:\underline{d}_{\mu_\phi}(y)\le 1/\kappa\}=D_{\mu_\phi}(1/\kappa). \end{align*} $$

This clearly yields the lemma.

Choose $ b\in (1/\kappa , a) $ . Put $ A_n:=\{y:\mu _\phi (B(y, r))<r^b\text { for all }r<2^{-n}\} $ . By the definition of $ \underline d_{\mu _\phi }(y) $ , we have

$$ \begin{align*}\{y:\underline d_{\mu_\phi}(y)>a\}\subset\bigcup_{n=1}^\infty A_n.\end{align*} $$

Thus, equation (5.7) is reduced to showing that for any $ n\ge 1 $ ,

(5.8) $$ \begin{align} \operatorname{\mathrm{\dim_H}} (\mathcal U^\kappa(x)\cap A_n)=0\quad\text{for }\mu_\phi\text{-}\text{a.e.}\,x. \end{align} $$

Step 2. The next objective is to prove equation (5.8).

Fix $ n\ge 1 $ . Let $ \epsilon>0 $ be arbitrary. Choose a large integer $ l\ge n $ with

(5.9) $$ \begin{align} 12\times 2^{-\kappa l}<2^{-n}\quad\text{and}\quad \gamma^312^b2^{(1-b\kappa)l}< \epsilon, \end{align} $$

where the constant $ \gamma $ is defined in equation (3.1). Let $ \theta _j=[\kappa j\log _{L_1}2]+1 $ , where $ L_1 $ is given in equation (2.2). Then, by equation (2.2), the length of each basic interval of generation $ \theta _j $ is smaller than $ 2^{-\kappa j} $ . Recall that $ \mathcal E $ is the set of endpoints of basic intervals which is countable. For any $ x\in [0,1]\setminus \mathcal E $ , define

$$ \begin{align*}\mathcal I_{j}(x):=\bigcup_{J\in\Sigma_{\theta_j}\colon d(I_{\theta_j}(x),J)<2^{-\kappa j}}J,\end{align*} $$

where $ d(\cdot ,\cdot ) $ is the Euclidean metric. Clearly, $ \mathcal I_j(x) $ covers the ball $ B(x, 2^{-\kappa j}) $ and is contained in $ B(x, 3\times 2^{-\kappa j}) $ . Moreover, if $ I_{\theta _j}(x)=I_{\theta _j}(y) $ , then $ \mathcal I_j(x)=\mathcal I_j(y) $ . With the notation $ \mathcal I_j(x) $ , we consider the set

$$ \begin{align*}G_{l,i}(x)= A_n\cap\bigg(\bigcap_{j=l}^i\bigcup_{k=1}^{2^j}\mathcal{I}_{j}(T^kx)\bigg).\end{align*} $$

The advantage of using $ \mathcal I_j(x) $ rather than $ B(x, 2^{-\kappa j}) $ is that the map $ x\mapsto G_{l,i}(x) $ is constant on each basic interval of generation $ 2^i+\theta _i $ . We are going to construct inductively a cover of $ G_{l,i}(x) $ by the family $ \{B(T^kx, 3\times 2^{-\kappa i}):k\in S_i(x)\} $ of balls, where $ S_i(x)\subset \{1,2,\ldots ,2^i\} $ .

For $ i=l $ , we let $ S_l(x)\subset \{1,2,\ldots , 2^l\} $ consist of those $ k\le 2^l $ such that $ \mathcal I_l(T^kx) $ intersects $ A_n $ . Suppose now that $ S_i(x) $ has been defined. We define $ S_{i+1}(x) $ to consist of those $ k\le 2^{i+1} $ such that $ \mathcal I_{i+1}(T^kx) $ intersects $ G_{l,i}(x) $ . Then the family $ \{B(T^kx, 3\times 2^{-\kappa (i+1)}):k\in S_{i+1}(x)\} $ of balls forms a cover of $ G_{l,i+1}(x) $ , and the construction is completed.

With the aid of the notation $ \mathcal I_{i+1}(T^kx) $ , one can verify that $ x\mapsto S_{i+1}(x) $ is constant on each basic interval of generation $ 2^{i+1}+\theta _{i+1} $ . Let $ N_{i+1}(x):=\sharp S_{i+1}(x) $ , then $ x\mapsto N_{i+1}(x) $ is also constant on each basic interval of generation $ 2^{i+1}+\theta _{i+1} $ .

To establish equation (5.8), we need to estimate $ N_{i+1}(x) $ . For those $ k\in S_{i+1}(x)\cap \{1,2,\ldots ,2^i\} $ , since $ \mathcal I_{i+1}(T^kx)\subset \mathcal I_i(T^kx) $ and $ G_{l,i}(x)\subset G_{l,i-1}(x) $ , we must have that $ \mathcal I_i(T^kx) $ intersects $ G_{l,i-1}(x) $ , and hence $ k\in S_i(x) $ . However, since $ \mathcal I_{i+1}(T^kx) $ is contained in $ B(T^kx,3\times 2^{-\kappa (i+1)}) $ , if $ \mathcal I_{i+1}(T^kx) $ has non-empty intersection with $ G_{l,i}(x) $ , then the distance between $ T^kx $ and $ G_{l,i}(x) $ is less than $ 3\times 2^{-\kappa (i+1)} $ . In particular,

(5.10) $$ \begin{align} T^kx \in\{y: d(y, G_{l,i}(x))<3\times2^{-\kappa(i+1)}\}\subset \bigcup_{J\in\Sigma_{\theta_i}\colon d(J,G_{l,i}(x))<3\times2^{-\kappa (i+1)}}J. \end{align} $$

Denote the right-hand side union as $ \hat G_{l,i}(x) $ . The set $ \hat G_{l,i}(x) $ is nothing but the union of cylinders of level $ \theta _i $ whose distance from $ G_{l,i}(x) $ is less than $ 3\times 2^{-\kappa (i+1)} $ . Thus, by the fact that $ x\mapsto G_{l,i}(x) $ is constant on each basic interval of generation $ 2^i+\theta _i $ , we have

(5.11) $$ \begin{align} \hat G_{l,i}(x)=\hat G_{l,i}(y),\quad \text{whenever } I_{2^i+\theta_i}(x)=I_{2^i+\theta_i}(y). \end{align} $$

According to the above discussion, it holds that

$$ \begin{align*}N_{i+1}(x)\le N_i(x)+M_{i+1}(x),\end{align*} $$

where $ M_{i+1}(x) $ is the number of $ 2^i<k\le 2^{i+1} $ for which $ T^k x $ intersects $ \hat G_{l,i}(x) $ .

The function $ M_{i+1}(x) $ can further be written as

$$ \begin{align*} M_{i+1}(x)=\sum_{k=2^i+1}^{2^{i+1}}\chi_{\hat G_{l,i}(x)}(T^kx)=\sum_{k=2^i+1}^{2^i+\theta_i}\chi_{\hat G_{l,i}(x)}(T^kx)+\sum_{k=2^i+\theta_i+1}^{2^{i+1}}\chi_{\hat G_{l,i}(x)}(T^kx). \end{align*} $$

Since the countable set $ \mathcal E $ has zero $\mu _\phi $ -measure, all the functions given above are well defined for $ \mu _\phi $ -a.e. $ x $ . It follows from the locally constant property in equation (5.11) of $ \hat G_{l,i(x)} $ that

(5.12) $$ \begin{align} \notag\int M_{i+1}(x) d\mu_\phi(x)&=\!\sum_{k=2^i+1}^{2^i+\theta_i}\int\! \chi_{\hat G_{l,i}(x)}(T^kx)d\mu_\phi(x)+\!\sum_{k=2^i+\theta_i+1}^{2^{i+1}}\int\! \chi_{\hat G_{l,i}(x)}(T^kx)d\mu_\phi(x)\\ &\le \theta_i+\sum_{J\in\Sigma_{2^i+\theta_i}}\sum_{k=2^i+\theta_i+1}^{2^{i+1}}\int \chi_J(x) \chi_{\hat G_{l,i}(x)}(T^kx)d\mu_\phi(x)\notag\\ &=\theta_i+\sum_{J\in\Sigma_{2^i+\theta_i}}\sum_{k=2^i+\theta_i+1}^{2^{i+1}}\int \chi_J(x) \chi_{\hat G_{l,i}(x_J)}(T^kx)d\mu_\phi(x), \end{align} $$

where $ x_J $ is any fixed point of $ J $ . Now the task is to deal with the right-hand side summation. We deduce from the quasi-Bernoulli property in equation (3.2) of $ \mu _\phi $ that for each $ k>2^i+\theta _i $ ,

(5.13) $$ \begin{align} \int \chi_J(x) \chi_{\hat G_{l,i}(x_J)}(T^kx)d\mu_\phi(x)=\mu_\phi(J\cap T^{-k}(\hat G_{l,i}(x_J)))\le \gamma^3\mu_\phi(J)\mu_\phi(\hat{G}_{l,i}(x_J)). \end{align} $$

Since $ G_{l,i}(x_J) $ can be covered by the family $ \{B(T^kx_J, 3\times 2^{-\kappa i}):k\in S_i(x_J)\} $ of balls, then by equation (5.10), the family $ \mathcal F_i(x_J):=\{B(T^kx_J, 6\times 2^{-\kappa i}):k\in S_i(x_J)\} $ of enlarged balls forms a cover of $ \hat {G}_{l,i}(x_J) $ . Observe that each enlarged ball $ B\in \mathcal F_i(x_J) $ intersects $ A_n $ , and thus $ B\subset B(y,12\times 2^{-\kappa i}) $ for some $ y\in A_n $ . Then, by the definition of $ A_n $ and equation (5.9),

$$ \begin{align*}\mu_\phi(B)\le \mu_\phi(B(y,12\times 2^{-\kappa i}))\le12^b2^{-b\kappa i}.\end{align*} $$

Accordingly,

(5.14) $$ \begin{align} \mu_\phi(\hat{G}_{l,i}(x_J))\le 12^b2^{-b\kappa i}N_i(x_J). \end{align} $$

Recall that $ x\mapsto N_i(x) $ is constant on each basic interval of generation $ 2^i+\theta _i $ . Applying the upper bound of equation (5.14) on $ \mu _\phi (\hat {G}_{l,i}(x_J)) $ to equation (5.13), and then substituting equation (5.13) into equation (5.12), we have

$$ \begin{align*} \notag\int M_{i+1}(x) \,d\mu_\phi(x) &\le \theta_i+\sum_{J\in\Sigma_{2^i+\theta_i}}\sum_{k=2^i+\theta_i+1}^{2^{i+1}}\gamma^3\mu_\phi(\hat{G}_{l,i}(x_J))\mu_\phi(J)\\&\le \theta_i+\sum_{J\in\Sigma_{2^i+\theta_i}}\sum_{k=2^i+\theta_i+1}^{2^{i+1}}\gamma^312^b2^{-b\kappa i}N_i(x_J)\mu_\phi(J)\\&= \theta_i+\gamma^312^b2^{-b\kappa i}(2^i-\theta_i)\int N_i(x)\,d\mu_\phi(x)\\&\le \theta_i+\epsilon\int N_i(x)\,d\mu_\phi(x), \end{align*} $$

where the last inequality follows from equation (5.9). Since $ N_{i+1}(x)\le N_i(x)+M_{i+1}(x) $ , we have

(5.15) $$ \begin{align} \int N_{i+1}(x)\,d\mu_\phi(x)\le \theta_i+(1+\epsilon)\int N_i(x)\,d\mu_\phi(x). \end{align} $$

Note that equation (5.15) holds for all $ i\ge l $ and $ N_l(x)\le 2^l $ . Then,

$$ \begin{align*}\begin{split} \int N_{i+1}(x)\,d\mu_\phi(x)&\le \sum_{k=l}^{i}(1+\epsilon)^{i-k}\theta_k+(1+\epsilon)^{i-l+1}\int N_l(x)\,d\mu_\phi(x)\\ &<(1+\epsilon)^i(i\theta_i+2^l). \end{split}\end{align*} $$

By Markov’s inequality,

$$ \begin{align*} &\mu_\phi(\{x:N_{i+1}(x)\ge (1+\epsilon)^{2i}(i\theta_i+2^l)\})\\&\quad\le \mu_\phi\bigg(\bigg\{x:N_{i+1}(x)\ge (1+\epsilon)^i\int N_{i+1}(x)\,d\mu_\phi(x)\bigg\}\bigg)\\ &\quad\le (1+\epsilon)^{-i}, \end{align*} $$

which is summable over $ i $ . Hence, for $ \mu _\phi $ -a.e. $ x $ , there is an $ i_0(x) $ such that

(5.16) $$ \begin{align} N_{i+1}(x)\le (1+\epsilon)^{2i}(i\theta_i+2^l) \end{align} $$

holds for all $ i\ge i_0(x) $ .

Denote by $ F_{l,\epsilon } $ the full measure set on which equation (5.16) holds. Let $ x\in F_{l,\epsilon } $ . Then, such an $ i_0(x) $ exists. With $ i\ge i_0(x) $ , we may cover the set

$$ \begin{align*}G_l(x)=A_n\cap\bigg(\bigcap_{j=l}^\infty\bigcup_{k=1}^{2^j}\mathcal I_j(T^kx)\bigg)\end{align*} $$

by $ N_i(x) $ balls of radius $ 3\times 2^{-\kappa i} $ . Since $ \theta _i=[\kappa i\log _{L_1}2]+1\le ci $ for some $ c>0 $ , we have

(5.17) $$ \begin{align} \operatorname{\mathrm{\dim_H}} G_l(x)\le\limsup_{i\to\infty}\frac{\log N_i(x)}{\log 2^{\kappa i}}\le\limsup_{i\to\infty}\frac{\log ((1+\epsilon)^{2i}(i\theta_i+2^l))}{\log 2^{\kappa i}}=\frac{2\log (1+\epsilon)}{\kappa\log2}. \end{align} $$

Let $ (\epsilon _m)_{m\ge 1} $ be a monotonically decreasing sequence of real numbers converging to $ 0 $ . For each $ \epsilon _m $ , choose an integer $ l_m $ satisfying equation (5.9), but with $ \epsilon $ replaced by $ \epsilon _m $ . For every $ m\ge 1 $ , by the same reason as equation (5.17), there exists a full $ \mu _\phi $ -measure set $ F_{l_m,\epsilon _m} $ such that

$$ \begin{align*}\text{for all }x\in F_{l_m,\epsilon_m},\quad\operatorname{\mathrm{\dim_H}} G_{l_m}(x)\le\frac{2\log (1+\epsilon_m)}{\kappa\log2}.\end{align*} $$

By taking the intersection of the countable full $ \mu _\phi $ -measure sets $ (F_{l_m,\epsilon _m})_{m\ge 1} $ , and using the fact that $ G_l(x) $ is increasing in $ l $ , we obtain that for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*}\text{for any }l\ge 1,\quad\operatorname{\mathrm{\dim_H}} G_l(x)\le\lim_{m\to \infty}\operatorname{\mathrm{\dim_H}} G_{l_m}(x)=0.\end{align*} $$

We conclude equation (5.8) by noting that

$$ \begin{align*}\mathcal U^\kappa(x)\cap A_n\subset\bigcup_{l\ge 1}G_l(x).\end{align*} $$

Remark 8. Recall that Lemma 4.1 exhibits a relation between $ \mathcal U^\kappa (x) $ and hitting time:

$$ \begin{align*} \{ y:\overline{R}(x,y)<1/\kappa\}\subset\mathcal U^\kappa(x)\subset\{ y:\overline{R}(x,y)\le1/\kappa\}.\end{align*} $$

With this relation in mind, it is natural to investigate the size of the level sets

$$ \begin{align*}\{y:R(x,y)=1/\kappa\},\quad \kappa\in(0,\infty).\end{align*} $$

Lemmas 5.4 and 5.7 together with the inclusions

$$ \begin{align*}&\{y:R(x,y)=1/\kappa\}\subset\{y:\overline{R}(x,y)\le 1/\kappa\}\quad\text{and}\\ &\quad\{y:R(x,y)=1/\kappa\}\subset\{y:\overline{R}(x,y)\ge 1/\kappa\}\end{align*} $$

give the upper bound for Hausdorff dimension:

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\{y:R(x,y)=1/\kappa\}\le D_{\mu_\phi}(1/\kappa)\quad\text{for }\mu_\phi\text{-}\text{a.e.}\, x. \end{align*} $$

The lower bound, coinciding with the upper bound, can be proved by the same argument as that of Lemma 5.1. Thus for $ \mu _\phi $ -a.e. $ x $ ,

$$ \begin{align*}\operatorname{\mathrm{\dim_H}}\{y:R(x,y)=1/\kappa\}=D_{\mu_\phi}(1/\kappa)\quad\text{if }1/\kappa\notin \{\alpha_-,\alpha_+\}. \end{align*} $$

Acknowledgements

The authors would like to thank Tomas Persson for pointing out that Lebesgue measure statement stated in Example 1 can be concluded from Ref. [Reference Holland, Kirsebom, Kunde and Persson15] by a Fubini-based argument. The authors are very grateful to the anonymous referee for his/her valuable comments and suggestions, especially for pointing out that item (2) of Theorem 1.2 is valid in a wider setting and is a direct consequence of [Reference Holland, Kirsebom, Kunde and Persson15, Theorem 3.2] and [Reference Kirsebom, Kunde and Persson19, Proposition 1 and Theorem 5].

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Figure 0

Figure 1 The multifractal spectrum of $ \mu _\phi $ and the maps $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal B^\kappa (x) $, $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x) $, and $ 1/\kappa \mapsto \operatorname {\mathrm {\dim _H}}\mathcal L^\kappa (x) $.