Dimension estimates for $C^1$ iterated function systems and repellers. Part I

Abstract

This is the first paper in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.

1 Introduction

The dimension theory of iterated function systems (IFSs) and dynamical repellers has developed into an important field of research during the last 40 years. One of the main objectives is to estimate variant notions of dimension of the invariant sets and measures involved. Despite many new and significant developments in recent years, only the cases of conformal repellers and attractors of conformal IFSs under a certain separation condition have been completely understood. In such cases, the topological pressure plays a crucial role in the theory. Indeed, the Hausdorff and box-counting dimensions of the repeller X for a $C^1$ conformal expanding map f are given by the unique number s satisfying the Bowen–Ruelle formula $P(X, f, -s\log \|D_xf\|)=0$ , where the functional P is the topological pressure; see [8, 23, 38]. A similar formula is obtained for the Hausdorff and box-counting dimensions of the attractor of a conformal IFS satisfying the open set condition (see, for example, [34]).

The study of dimension in the non-conformal cases has proved to be much more difficult. In his seminal paper [13], Falconer established a general upper bound on the Hausdorff and box-counting dimensions of a self-affine set (which is the attractor of an IFS consisting of contracting affine maps) in terms of the so-called affinity dimension, and proved that for typical self-affine sets under a mild assumption this upper bound is equal to the dimension. So far substantial progress has been made towards understanding when the Hausdorff and box-counting dimensions of a concrete planar self-affine set are equal to its affinity dimension; see [3, 24] and the references therein. However, very little has been known in the higher-dimensional case.

In [15, Theorem 5.3], by developing a subadditive version of the thermodynamical formalism, Falconer showed that the upper box-counting dimension of a mixing repeller $\Lambda $ of a non-conformal $C^2$ mapping $\psi $ , under the distortion condition

(1.1) $$ \begin{align} \|(D_x\psi)^{-1}\|^2 \|D_x\psi\|<1 \quad \mbox{for }x\in \Lambda, \end{align} $$

is bounded above by the zero point of the (subadditive) topological pressure associated with the singular value functions of the derivatives of iterates of $\psi $ . We write this zero point as $\dim _{S^*}\Lambda $ and call it the singularity dimension of $\Lambda $ (see Definition 6.1 for the details). Condition (1.1) is used to prove the bounded distortion property of the singular value functions, which enables one to control the distortion of balls after many iterations (see [15, Lemma 5.2]). Examples involving ‘triangular maps’ were constructed in [32] to show that the condition (1.1) is necessary for this bounded distortion property.

Using a quite different approach, Zhang [40] proved that the Hausdorff dimension of the repeller of an arbitrary $C^1$ expanding map $\psi $ is also bounded above by the singularity dimension. (We remark that the upper bound given by Zhang was defined in a slightly different way, but it is equal to the singularity dimension; see [2, Corollary 2] for a proof.) The basic technique used in Zhang’s proof is to estimate the Hausdorff measure of $\psi (A)$ for small sets A (see [40, Lemma 3]), which is applied to only one iteration so it avoids assuming any distortion condition. However, his method does not apply to the box-counting dimension.

Thanks to the results of Falconer and Zhang, a natural question arises as to whether the upper box-counting dimension of a $C^1$ repeller is always bounded above by its singularity dimension. The challenge here is the lack of valid tools to analyze the geometry of the images of balls under a large number of iterations of $C^1$ maps. In [4, Theorem 3], Barreira made a positive claim concerning this question, but his proof contains a crucial mistake ^† , as found by Manning and Simon [32]. In a recent paper [9, Theorem 3.2], Cao, Pesin and Zhao obtained an upper bound for the upper box-counting dimension of the repellers of $C^{1+\alpha }$ maps satisfying a certain dominated splitting property. However, that upper bound depends on the splitting involved and is usually strictly larger than the singularity dimension.

In the present paper, we give an affirmative answer to the above question. We also establish an analogous result for the attractors of $C^1$ non-conformal IFSs. Meanwhile we prove that the upper packing dimension of an ergodic invariant measure supported on a $C^1$ repeller (respectively, the projection of an ergodic measure on the attractor of a $C^1$ IFS) is bounded above by its Lyapunov dimension.

In the continuation of this paper [22] we verify that upper bound estimates of the dimensions of the attractors and ergodic measures in the previous paragraph give the exact values of the dimensions for some families of $C^1$ non-conformal IFSs in ${\Bbb R}^d$ at least typically. ‘Typically’ means that the assertions hold for almost all translations of the system. These families include the $C^1$ non-conformal IFSs in ${\Bbb R}^d$ for which all differentials are either diagonal matrices, or all differentials are lower triangular matrices and satisfy a certain domination condition.

We first state our results for $C^1$ IFSs. To this end, let us introduce some notation and definitions. Let Z be a compact subset of ${\Bbb R}^d$ . A finite family $\{f_i\}_{i=1}^\ell $ of contracting self-maps on Z is called a $C^1$ iterated function system, if there exists an open set $U\supset Z$ such that each $f_i$ extends to a contracting $C^1$ -diffeomorphism $f_i: U\to f_i(U)\subset U$ . Let K be the attractor of the IFS, that is, K is the unique non-empty compact subset of ${\Bbb R}^d$ such that

(1.2) $$ \begin{align} K=\bigcup_{i=1}^\ell f_i(K) \end{align} $$

(cf. [16]).

Let $(\Sigma ,\sigma )$ be the one-sided full shift over the alphabet $\{1,\ldots , \ell \}$ (cf. [7]). Let $\Pi : \Sigma \to K$ denote the canonical coding map associated with the IFS $\{f_i\}_{i=1}^\ell $ . That is,

(1.3) $$ \begin{align} \Pi(x)=\lim_{n\to \infty} f_{x_1}\circ \cdots\circ f_{x_n}(z), \quad x=(x_n)_{n=1}^\infty, \end{align} $$

with $z\in U$ . The definition of $\Pi $ is independent of the choice of z.

For any compact subset X of $\Sigma $ with $\sigma X\subset X$ , we call $(X,\sigma )$ a one-sided subshift or simply a subshift over $\{1,\ldots , \ell \}$ and let $\dim _SX$ denote the singularity dimension of X with respect to the IFS $\{f_i\}_{i=1}^\ell $ (cf. Definition 2.5).

For a set $E\subset {\Bbb R}^d$ , let $\overline {\dim }_BE$ denote the upper box-counting dimension of E (cf. [16]). The first result in this paper is the following theorem, stating that the upper box-counting dimension of $\Pi (X)$ is bounded above by the singularity dimension of X.

Theorem 1.1. Let $X\subset \Sigma $ be compact and $\sigma X\subset X$ . Then $\overline {\dim }_B\Pi (X)\leq \dim _SX$ . In particular,

$$ \begin{align*}\overline{\dim}_BK\leq \dim_S\Sigma.\end{align*} $$

For an ergodic $\sigma $ -invariant measure m on $\Sigma $ , let $\dim _{L}m$ denote the Lyapunov dimension of m with respect to $\{f_i\}_{i=1}^\ell $ (cf. Definition 2.6). For a Borel probability measure $\eta $ on ${\Bbb R}^d$ (or a manifold), let $\overline {\dim }_P\eta $ denote the upper packing dimension of $\eta $ . That is,

$$ \begin{align*} \overline{\dim}_P\eta={\textrm{esssup}}_{x\in {\textrm{supp}}(\eta)}\overline{d}(\eta,x), \quad \mbox{with }\overline{d}(\eta, x):=\limsup_{r\to 0}\frac{\log \eta(B(x,r))}{\log r}, \end{align*} $$

where $B(x,r)$ denotes the closed ball centered at x of radius r. Equivalently,

$$ \begin{align*}\overline{\dim}_P\eta=\inf\{ \dim_PF: F\ \mbox{is a Borel set with } \eta(F)=1\}, \end{align*} $$

where $\dim _PF$ stands for the packing dimension of F (cf. [16]). See, for example, [17] for a proof.

Our second result can be viewed as a measure analogue of Theorem 1.1.

Theorem 1.2. Let m be an ergodic $\sigma $ -invariant measure on $\Sigma $ . Then

$$ \begin{align*}\overline{\dim}_P(m\circ \Pi^{-1})\leq \dim_{L}m,\end{align*} $$

where $m\circ \Pi ^{-1}$ stands for the push-forward of m by $\Pi $ .

The above theorem improves a result of Jordan and Pollicott [26, Theorem 1] which states that

$$ \begin{align*}\overline{\dim}_H(m\circ \Pi^{-1})\leq \dim_{L}m\end{align*} $$

under a slightly more general setting, where $\overline {\dim }_H(m\circ \Pi ^{-1})$ stands for the upper Hausdorff dimension of $m\circ \Pi ^{-1}$ . Recall that the upper Hausdorff dimension of a measure is the infimum of the Hausdorff dimension of Borel sets of full measure, which is always less than or equal to the upper packing dimension of the measure. It is worth pointing out that Theorem 1.2 was proved previously by Jordan [25] and Rossi [36] in the special case when $\{f_i\}_{i=1}^\ell $ is an affine IFS.

Next, we turn to the case of repellers. Let ${\pmb M}$ be a smooth Riemannian manifold of dimension d and $\psi : {\pmb M}\to {\pmb M}$ a $C^1$ -map. Let $\Lambda $ be a compact subset of ${\pmb M}$ such that $\psi (\Lambda )=\Lambda $ . We say that $\psi $ is expanding on the repeller $\Lambda $ if:

1. (a) there exists $\unicode{x3bb}>1$ such that $\|(D_z\psi ) v\|\geq \unicode{x3bb} \|v\|$ for all $z\in \Lambda $ , $v\in T_z{\pmb M}$ (with respect to a Riemannian metric on ${\pmb M}$ );

2. (b) there exists an open neighborhood V of $\Lambda $ such that

   $$ \begin{align*} \Lambda=\{z\in V: \psi^n(z)\in V \mbox{ for all } n\geq 0\}. \end{align*} $$

We refer the reader to [35, §20] for more details. In what follows we always assume that $\Lambda $ is a repeller of $\psi $ . Let $\dim _{S^*}\Lambda $ denote the singular dimension of $\Lambda $ with respect to $\psi $ (see Definition 6.1). For an ergodic $\psi $ -invariant measure $\mu $ on $\Lambda $ , let $\dim _{L^*}\mu $ be the Lyapunov dimension of $\mu $ with respect to $\psi $ (see Definition 6.2). Analogously to Theorems 1.1–1.2, we have the following results.

Theorem 1.3. Let $\Lambda $ be the repeller of $\psi $ . Then

$$ \begin{align*}\overline{\dim}_B \Lambda\leq \dim_{S^*}\Lambda.\end{align*} $$

Theorem 1.4. Let $\mu $ be an ergodic $\psi $ -invariant measure supported on $\Lambda $ . Then

$$ \begin{align*} \overline{\dim}_P\mu\leq \dim_{L^*}\mu. \end{align*} $$

For the estimates of the box-counting dimension of attractors of $C^1$ non-conformal IFSs (respectively, $C^1$ repellers), the reader may reasonably ask what difficulties arose in the previous work [15] which the present paper overcomes. Below we give an explanation and roughly illustrate our strategy for the proof.

Let us give an account of the IFS case. The case of repellers is similar. Let K be the attractor of a $C^1$ IFS $\{f_i\}_{i=1}^\ell $ . To estimate $\overline {\dim }_BK$ , by definition one needs to estimate for given $r>0$ the smallest number of balls of radius r required to cover K, say, $N_r(K)$ . To this end, one may iterate the IFS to get $K=\bigcup _{i_1\ldots i_n} f_{i_1\ldots i_n}(K)$ and then estimate $N_r(f_{i_1\ldots i_n}(K))$ separately, where $f_{i_1\ldots i_n}:=f_{i_1}\circ \cdots \circ f_{i_n}$ . For this purpose, one needs to estimate $N_r(f_{i_1\ldots i_n}(B))$ , where B is a fixed ball covering K.

Under the strong assumption of distortion property, Falconer was able to show that $f_{i_1\ldots i_n}(B)$ is roughly comparable to the ellipsoid $(D_x f_{i_1\ldots i_n})(B)$ for each $x\in B$ (see [15, Lemma 5.2]); then, by cutting the ellipsoid into roughly round pieces, he could use a certain singular value function to give an upper bound of $N_r(f_{i_1\ldots i_n}(B))$ , and then apply the subadditive thermodynamic formalism to estimate the growth rate of $\sum _{i_1\ldots i_n}N_r(f_{i_1\ldots i_n}(B))$ . However, in the general $C^1$ non-conformal case, this approach is no longer feasible, since it seems hopeless to analyze the geometric shape of $f_{i_1\ldots i_n}(B)$ when n is large.

The strategy of our approach is quite different. We use an observation going back to Douady and Oesterlé [12] (see also [40]) that, for a given $C^1$ map f, when $B_0$ is a small enough ball in a fixed bounded region, $f(B_0)$ is close to being an ellipsoid and so can be covered by a certain number of balls controlled by the singular values of the differentials of f (see Lemma 4.1). Since the maps $f_i$ in the IFS are contracting, we may apply this fact to the maps $f_{i_n}$ , $f_{i_{n-1}}$ , …, $f_{i_1}$ recursively. Roughly speaking, suppose that $B_0$ is a ball of small radius $r_0$ . Then $f_{i_n}(B_0)$ can be covered by $N_1$ balls of radius $r_1$ , and the image of each of them under $f_{i_{n-1}}$ can be covered by $N_2$ balls of radius of $r_2$ , and so on, where $N_j, r_j/r_{j-1}$ ( $j=1,\ldots , n$ ) can be controlled by the singular values of the differentials of $f_{i_{n-j+1}}$ . In this way we get an estimate that $N_{r_n}(f_{i_1\ldots i_n}(B_0))\leq N_1\ldots N_n$ (see Proposition 4.2 for a more precise statement), which is in spirit analogous to the corresponding estimate for the Hausdorff measure by Zhang [40]. In this process, we do not need to consider the differentials of $f_{i_1\ldots i_n}$ and so no distortion property is required. By developing a key technique from the thermodynamic formalism (see Proposition 3.4), we can get an upper bound for $\overline {\dim }_BK$ , say $s_1$ . Replacing the IFS $\{f_i\}_{i=1}^\ell $ by its nth iteration $\{f_{i_1\ldots i_n}\}$ , we get other upper bounds $s_n$ . Again using a technique in the thermodynamic formalism (see Proposition 3.1), we manage to show that $\lim \inf s_n$ is bounded above by the singularity dimension.

The proof of Theorem 1.2 is also based on the above strategy. For an ergodic measure m on $\Sigma $ , we are able, using the above covering arguments and ergodic theorems, to provide sharp estimates on the growth rates of $N_{(u_k)^n}(f_{\textbf {i|n}}(K))$ for m-a.e. (almost every) ${\textbf {i}}\in \Sigma $ , where $u_k=\exp (\unicode{x3bb} _k)$ , $k=1,\ldots , d$ , with $\unicode{x3bb} _k$ being the kth Lyapunov exponent of the matrix cocycle $A({\textbf {i}}, n):=D_{\pi \sigma ^n{\textbf {i}}}f_{{\textbf {i}}|n}$ with respect to m. More precisely, we have the following inequality (see Lemma 5.1):

$$ \begin{align*} \lim_{n\to \infty} \frac{1}{n} \log( N_{(u_k)^n}(f_{\textbf{i|n}}(K)))\leq (\unicode{x3bb}_1+\cdots +\unicode{x3bb}_{k-1})-(k-1)\unicode{x3bb}_k \end{align*} $$

for m-a.e. ${\textbf {i}}\in \Sigma $ , with the convention that $\unicode{x3bb} _1+\cdots +\unicode{x3bb} _{k-1}=0$ if $k=1$ . The proof of Lemma 5.1 is delicate. In particular, we need to apply a special version of Kingman’s subadditive ergodic theorem which is stated and proved in Lemma 2.8. Theorem 1.2 is then derived from Lemma 5.1 by using an idea employed in [25, 36].

This paper is organized as follows. In §2 we give some preliminaries about the subadditive thermodynamic formalism and give the definitions of the singularity and Lyapunov dimensions with respect to a $C^1$ IFS. In §3 we prove two auxiliary results (Propositions 3.1 and 3.4) which play a key role in the proof of Theorem 1.1 (and of Theorem 1.3). The proofs of Theorems 1.1–1.2 are given in §§4–5, respectively. In §6 we give the definitions of the singularity and Lyapunov dimensions in the repeller case and prove Theorems 1.3–1.4. For the convenience of the reader, in the Appendix we summarize the main notation and typographical conventions used in this paper.

2 Preliminaries

2.1 Variational principle for subadditive pressure

In order to define the singularity and Lyapunov dimensions and prove our main results, we require some elements from the subadditive thermodynamic formalism.

Let $(X,d)$ be a compact metric space and $T:X\to X$ a continuous mapping. We call $(X,T)$ a topological dynamical system. For $x,y\in X$ and $n\in {\Bbb N}$ , we define

(2.1) $$ \begin{align} d_n(x,y):=\max\limits_{0 \leq i \leq n-1}d(T^i(x),T^i(y)). \end{align} $$

A set $E \subset X$ is called $(n,\varepsilon )$ -separated if for every distinct $x,y\in E$ we have $d_n(x,y)>\varepsilon $ .

Let $C(X)$ denote the set of real-valued continuous functions on X. Let $\mathcal {G}=\{g_n\}_{n=1}^{\infty }$ be a subadditive potential on X, that is, $g_n\in C(X)$ for all $n\geq 1$ such that

(2.2) $$ \begin{align} g_{m+n}(x) \leq g_{n}(x)+g_{m}(T^nx) \quad\mbox{for all } x\in X \mbox{ and } n,m\in {\Bbb N}. \end{align} $$

Following [10], below we define the topological pressure of $\mathcal {G}$ .

Definition 2.1. For given $n\in \mathbb {N}$ and $\varepsilon>0$ , we define

(2.3) $$ \begin{align} P_n(X, T, \mathcal{G},\varepsilon):= \sup\bigg\{ \sum\limits_{x\in E} \exp (g_n(x)): E \mbox{ is an}\ (n,\varepsilon)\mbox{-separated set } \bigg\}. \end{align} $$

Then the topological pressure of $\mathcal {G}$ with respect to T is defined by

(2.4) $$ \begin{align} P(X, T, \mathcal{G}):= \lim\limits_{\varepsilon\to0} \limsup\limits_{n\to\infty} \frac{1}{n}\log P_n(X,T,\mathcal{G},\varepsilon). \end{align} $$

If the potential $\mathcal G$ is additive, that is, $g_n=S_ng:=\sum _{k=0}^{n-1}g\circ T^k$ for some $g\in C(X)$ , then $P(X, T, \mathcal G)$ recovers the classical topological pressure $P(X, T, g)$ of g (see, for example, [39]).

Let $\mathcal {M}(X)$ denote the set of Borel probability measures on X, and $\mathcal {M}(X,T)$ the set of T-invariant Borel probability measures on X. For $\mu \in \mathcal {M}(X,T)$ , let $h_\mu (T)$ denote the measure-theoretic entropy of $\mu $ with respect to T (cf. [39]). Moreover, for $\mu \in \mathcal M(X, T)$ , by subadditivity we have

(2.5) $$ \begin{align} \mathcal{G}_*(\mu):=\lim_{n\to\infty} \frac{1}{n} \int g_n \,d\mu=\inf_n \frac{1}{n} \int g_n \,d\mu\in [-\infty,\infty). \end{align} $$

See, for example, [39, Theorem 10.1]. We call $\mathcal {G}_*(\mu )$ the Lyapunov exponent of $\mathcal {G}$ with respect to $\mu $ .

The proofs of our main results rely on the following general variational principle for the topological pressure of subadditive potentials.

Theorem 2.2. [10, Theorem 1.1]

Let $\mathcal {G}=\{g_n\}_{n=1}^{\infty }$ be a subadditive potential on a topological dynamical system $(X,T)$ . Suppose that the topological entropy of T is finite. Then

(2.6) $$ \begin{align} P(X, T,\mathcal{G})= \sup\{h_\mu(T)+\mathcal{G}_*(\mu): \mu\in\mathcal{M}(X,T) \}. \end{align} $$

Particular cases of the above result, under stronger assumptions on the dynamical systems and the potentials, were previously obtained by many authors; see, for example, [5, 14, 18, 21, 28, 33] and references therein.

Measures that achieve the supremum in (2.6) are called equilibrium measures for the potential ${\mathcal G}$ . There exists at least one ergodic equilibrium measure when the entropy map $\mu \mapsto h_\mu (T)$ is upper semi-continuous; this is the case when $(X,T)$ is a subshift (see, for example, [19, Proposition 3.5] and the remark there).

The following well-known result is also needed in our proofs.

Lemma 2.3. Let $X_i, i= 1,2$ , be compact metric spaces and let $T_i: X_i \to X_i$ be continuous. Suppose $\pi : X_1 \to X_2$ is a continuous surjection such that the following diagram commutes:

Then $\pi _*:{\mathcal M}(X_1, T_1)\rightarrow {\mathcal M}(X_2, T_2)$ (defined by $\mu \mapsto \mu \circ \pi ^{-1}$ ) is surjective.

If, furthermore, there is an integer $q>0$ so that $\pi ^{-1}(y)$ has at most q elements for each $y\in X_2$ , then

$$ \begin{align*} h_{\mu}(T_1)=h_{\mu\circ \pi^{-1}}(T_2) \end{align*} $$

for each $\mu \in {\mathcal M}(X_1, T_1)$ .

Proof. The first part of the result is the same as [31, Ch. IV, Lemma 8.3]. The second part follows from the Abramov–Rokhlin formula (see [6]).

2.2 Subshifts

In this subsection we introduce some basic notation and definitions about subshifts.

Let $(\Sigma ,\sigma )$ be the one-sided full shift over the alphabet $\mathcal A=\{1,\ldots , \ell \}$ . That is, $\Sigma =\mathcal A^{\Bbb N}$ endowed with the product topology, and $\sigma :\Sigma \to \Sigma $ is the left shift defined by $(x_i)_{i=1}^\infty \mapsto (x_{i+1})_{i=1}^\infty $ . The topology of $\Sigma $ is compatible with the following metric on $\Sigma $ :

$$ \begin{align*} d(x,y)=2^{-\inf\{k: x_{k+1}\neq y_{k+1}\}} \quad \mbox{for }x=(x_i)_{i=1}^\infty, y=(y_i)_{i=1}^\infty. \end{align*} $$

For $x=(x_i)_{i=1}^\infty \in \Sigma $ and $n\in {\Bbb N}$ , write $x|n=x_1\ldots x_n$ .

Let X be a non-empty compact subset of $\Sigma $ satisfying $\sigma X\subset X$ . We call $(X, \sigma )$ a one-sided subshift or simply a subshift over $\mathcal A$ . We denote the collection of finite words allowed in X by $X^*$ , and the subset of $X^*$ of words of length n by $X_n^*$ . In particular, define, for $n\in {\Bbb N}$ ,

(2.7) $$ \begin{align} X^{(n)}:=\{(x_i)_{i=1}^\infty\in \mathcal A^{\Bbb N}: x_{kn+1}x_{kn+2}\ldots x_{(k+1)n}\in X_n^* \mbox { for all } k\geq 0\}. \end{align} $$

It is clear that $X\subset X^{(n)}$ for every $n\in {\Bbb N}$ .

Let $\mathcal G=\{ g_n\}_{n=1}^\infty $ be a subadditive potential on a subshift $(X,\sigma )$ . It is known that in such a case, the topological pressure of $\mathcal G$ can alternatively be defined by

(2.8) $$ \begin{align} P(X,\sigma, \mathcal G)= \lim_{n\to\infty} \frac{1}{n}\log\bigg( \sum_{\mathbf{i}\in X_n^*} \sup_{x\in [\mathbf{i}]\cap X} \exp(g_n(x)) \bigg), \end{align} $$

where $[\mathbf {i}]:=\{x\in \Sigma : x|n=\mathbf {i}\}$ for $\mathbf {i}\in \mathcal A^n$ ; see [10, p. 649]. The limit can be seen to exist by using a standard subadditivity argument. We remark that (2.8) was first introduced by Falconer in [14] for the definition of the topological pressure of subadditive potentials on a mixing repeller.

Below we provide a useful lemma.

Lemma 2.4. Let $({\mathcal A}^{\Bbb N},\sigma )$ be the one-sided full shift space over a finite alphabet ${\mathcal A}$ . Let $\nu \in {\mathcal M}({\mathcal A}^{\Bbb N}, \sigma ^m)$ for some $m\in {\Bbb N}$ . Set $\mu =({1}/{m})\sum _{k=0}^{m-1}\nu \circ \sigma ^{-k}$ . Then $\mu \in {\mathcal M}({\mathcal A}^{{\Bbb N}}, \sigma )$ and $h_\mu (\sigma )=({1}/{m})h_\nu (\sigma ^m)$ .

Proof. The lemma might be well known. However, we are not able to find a reference, so for the convenience of the reader we provide a self-contained proof. The $\sigma $ -invariance of $\mu $ follows directly from its definition, and we only need to prove that $h_\mu (\sigma )= ({1}/{m})h_\nu (\sigma ^m)$ .

Clearly, $\nu \circ \sigma ^{-k}\in {\mathcal M}({\mathcal A}^{\Bbb N}, \sigma ^m)$ for $k=0,\ldots , m-1$ . We claim that $h_{\nu \circ \sigma ^{-k}}(\sigma ^m)=h_{\nu }(\sigma ^m)$ for $k=1,\ldots , m-1$ . Without loss of generality we prove this in the case when $k=1$ . For $n\in {\Bbb N}$ , let ${\mathcal P}_n$ denote the partition of ${\mathcal A}^{\Bbb N}$ consisting of the nth cylinders of ${\mathcal A}^{\Bbb N}$ , that is, ${\mathcal P}_n=\{ [I]: I\in {\mathcal A}^n\}$ , and set $\sigma ^{-1} {\mathcal P}_n=\{\sigma ^{-1}([I]): I\in {\mathcal A}^n\}$ . Then it is direct to see that any element in ${\mathcal P}_n$ intersects at most $\#{\mathcal A}$ elements in $\sigma ^{-1}{\mathcal P}_n$ , and vice versa. Hence,

$$ \begin{align*} |H_\nu({\mathcal P}_n)-H_\nu(\sigma^{-1} {\mathcal P}_n)|\leq \log \#{\mathcal A}; \end{align*} $$

see, for example, [20, Lemma 4.6]. It follows that

$$ \begin{align*} h_{\nu\circ \sigma^{-1}} (\sigma^m)&=\lim_{n\to \infty} \frac{1}{n}H_{\nu\circ \sigma^{-1}} ({\mathcal P}_{nm})= \lim_{n\to \infty} \frac{1}{n}H_{\nu} ({\sigma^{-1}\mathcal P}_{nm})\\[6pt] &=\lim_{n\to \infty} \frac{1}{n}H_{\nu} ({\mathcal P}_{nm})=h_{\nu} (\sigma^m). \end{align*} $$

This proves the claim.

By the affinity of the measure-theoretic entropy $h_{(\cdot )}(\sigma ^m)$ (see [39, Theorem 8.1]), we have

$$ \begin{align*} h_\mu(\sigma^m)=\frac{1}{m}\sum_{k=0}^{m-1} h_{\nu\circ \sigma^{-k}}(\sigma^m)=h_\nu(\sigma^m), \end{align*} $$

where the second equality follows from the above claim. Hence, $h_\mu (\sigma )=({1}/{m})h_\mu (\sigma ^m)=({1}/{m})h_\nu (\sigma ^m)$ .

2.3 Singularity dimension and Lyapunov dimension with respect to $C^1$ IFSs

In this subsection, we define the singularity and Lyapunov dimensions with respect to $C^1$ IFSs. The corresponding definitions with respect to $C^1$ repellers will be given in §5.

Let $\{f_i\}_{i=1}^\ell $ be a $C^1$ IFS on ${\Bbb R}^d$ with attractor K. Let $(\Sigma ,\sigma )$ be the one-sided full shift over the alphabet $\{1,\ldots , \ell \}$ and let $\Pi : \Sigma \to K$ denote the corresponding coding map defined as in (1.3). For a differentiable function $f: U\subset {\Bbb R}^d\to {\Bbb R}^d$ , let $D_zf$ denote the differential of f at $z\in U$ .

For $T\in {\Bbb R}^{d\times d}$ , let $\alpha _1(T)\geq \cdots \geq \alpha _d(T)$ denote the singular values of T. Following [13], for $s\geq 0$ we define the singular value function $\phi ^s: {\Bbb R}^{d \times d}\to [0,\infty )$ as

(2.9) $$ \begin{align} \phi^s(T)=\begin{cases} \alpha_1(T)\cdots \alpha_k(T) \alpha_{k+1}^{s-k}(T) & \mbox{if }0\leq s\leq d,\\ \det(T)^{s/d} & \mbox{if } s>d, \end{cases} \end{align} $$

where $k=[s]$ is the integral part of s.

Definition 2.5. For a compact subset X of $\Sigma $ with $\sigma (X)\subset X$ , the singularity dimension of X with respect to $\{f_i\}_{i=1}^\ell $ , written as $\dim _SX$ , is the unique non-negative value s for which

$$ \begin{align*} P(X,\sigma, \mathcal G^s)=0, \end{align*} $$

where $\mathcal G^s=\{g_n^s\}_{n=1}^\infty $ is the subadditive potential on $\Sigma $ defined by

(2.10) $$ \begin{align} g_n^s(x)=\log \phi^s(D_{\Pi\sigma^n x}f_{x|n}), \quad x\in \Sigma, \end{align} $$

with $f_{x|n}:=f_{x_1}\circ \cdots \circ f_{x_n}$ for $x=(x_n)_{n=1}^\infty $ .

Definition 2.6. Let m be an ergodic $\sigma $ -invariant Borel probability measure on $\Sigma $ . For any $i\in \{1, \ldots ,d\}$ , the ith Lyapunov exponent of m is

(2.11) $$ \begin{align} \unicode{x3bb}_i(m):=\lim_{n\to \infty} \frac{1}{n}\int \log( \alpha_i(D_{\Pi\sigma^nx}f_{x|n}))\,dm(x). \end{align} $$

The Lyapunov dimension of m with respect to $\{f_i\}_{i=1}^\ell $ , written as $\dim _{L}m$ , is the unique non-negative value s for which

$$ \begin{align*} h_m(\sigma)+ \mathcal G^s_*(m)=0, \end{align*} $$

where $\mathcal G^s=\{g^s_n\}_{n=1}^\infty $ is defined as in (2.10) and $\mathcal G^s_*(m):=\lim _{n\to \infty } ({1}/{n})\int g^s_n\,dm$ . See Figure 1 for the mapping $s\mapsto -\mathcal G^s_*(m)$ in the case when $d=2$ .

Figure 1 The connection between Lyapunov dimension, entropy and the function $s\mapsto -\mathcal {G}^s_*(m)$ when $d=2$ .

It follows from the definition of the singular value function $\phi ^s$ that, for an ergodic measure m, we have

$$ \begin{align*} \mathcal G^s_*(m)= \begin{cases} \unicode{x3bb}_1(m)+\cdots+\unicode{x3bb}_{[s]}(m)+(s-[s])\unicode{x3bb}_{[s]+1}(m) & \text{if}\ s < d, \\ \frac{s}{d} (\unicode{x3bb}_1(m)+\cdots+\unicode{x3bb}_{d}(m) ) & \text{if}\ s \geq d. \end{cases} \end{align*} $$

Observe that, in the special case when all the Lyapunov exponents are equal to the same $\unicode{x3bb} $ , we have $\dim _{L}m={h_m(\sigma )}/{-\unicode{x3bb} }$ .

Remark 2.7

1. (i) The concept of singularity dimension was first introduced by Falconer [13, 15]; see also [29]. It is also called affinity dimension when the IFS $\{f_i\}_{i=1}^\ell $ is affine, that is, each map $f_i$ is affine.

2. (ii) The definition of Lyapunov dimension of ergodic measures with respect to an IFS presented above was taken from [26]. It is a generalization of that given in [27] for affine IFSs.

2.4 A special consequence of Kingman’s subadditive ergodic theorem

Here we state a special consequence of Kingman’s subadditive ergodic theorem which will be needed in the proof of Lemma 5.1.

Lemma 2.8. Let T be a measure-preserving transformation of the probability space $(X, {\mathcal B}, m)$ , and let $\{g_{n}\}_{{n\in {\mathbb {N}}}}$ be a sequence of $L^{1}$ functions satisfying the following subadditivity relation:

$$ \begin{align*}g_{{n+m}}(x)\leq g_{n}(x)+g_{m}(T^{n}x) \quad \mbox{for all }x\in X.\end{align*} $$

Suppose that there exists $C>0$ such that

(2.12) $$ \begin{align} |g_n(x)|\leq Cn \quad \mbox{for all}\ x\in X\ \mbox{and}\ n\in {\Bbb N}. \end{align} $$

Then

$$ \begin{align*} \lim _{n\to \infty }E\bigg(\frac {g_{n}}{n}\bigg|{\mathcal C}_n\bigg)(x)=g(x):=\lim_{n\to \infty}\frac{g_n(x)}{n} \end{align*} $$

for m-a.e. x, where

$$ \begin{align*}{\mathcal C}_n:=\{B\in {\mathcal B}: T^{-n}B =B \; a.e.\},\end{align*} $$

$E(\cdot |\cdot )$ denotes the conditional expectation and $g(x)$ is T-invariant.

Proof. By Kingman’s subadditive ergodic theorem, ${g_n}/{n}$ converges pointwise to a T-invariant function g-almost everywhere. Meanwhile, by Birkhoff’s ergodic theorem, for each $n\in {\Bbb N}$ ,

$$ \begin{align*} E(g_n|{\mathcal C}_n)(x)=\lim_{k\to \infty}\frac{1}{k}\sum_{j=0}^{k-1}g_n(T^{jn}x)\mbox{ almost everywhere}. \end{align*} $$

Since $\sum _{j=0}^{k-1}g_n(T^{jn}x)\geq g_{kn}(x)$ by subadditivity, it follows that

(2.13) $$ \begin{align} E\bigg(\frac {g_{n}}{n}\bigg|{\mathcal C}_n\bigg)(x)\geq \lim_{k\to \infty} \frac{g_{nk}(x)}{nk} = g(x)\mbox{ almost everywhere}. \end{align} $$

Since the sequence $\{g_n\}$ is subadditive and satisfies (2.12), by [28, Lemma 2.2], for any $0< k< n$ ,

$$ \begin{align*} \frac{g_n(x)}{n}\leq \frac{1}{kn}\sum_{j=0}^{n-1} g_k(T^jx)+\frac{3kC}{n},\quad x\in X. \end{align*} $$

As a consequence,

(2.14) $$ \begin{align} E\bigg(\frac {g_{n}}{n}\bigg|{\mathcal C}_n\bigg)(x)\leq \frac{1}{kn}E\bigg(\sum_{j=0}^{n-1}g_{k}\circ T^j\bigg|{\mathcal C}_n\bigg)(x)+\frac{3kC}{n}. \end{align} $$

Notice that, for each $f\in L^1$ and $n\in {\Bbb N}$ ,

(2.15) $$ \begin{align} E\bigg(\sum_{j=0}^{n-1}f\circ T^j\bigg|{\mathcal C}_n\bigg)=nE(f|{\mathcal C}_1) \mbox{ almost everywhere}. \end{align} $$

To see the above identity, one simply applies Birkhoff’s ergodic theorem (with respect to the transformations $T^n$ and T, respectively) to the following limits:

$$ \begin{align*} \lim_{k\to \infty}\frac{1}{k}\sum_{j=0}^{k-1}(f+f\circ T+\cdots+f\circ T^{n-1})(T^{nj}x) =\lim_{k\to \infty} \frac{1}{k}\sum_{j=0}^{nk-1}f(T^jx).\end{align*} $$

Now applying the identity (2.15) (with $f=g_k$ ) to (2.14) yields

$$ \begin{align*} E\bigg(\frac {g_{n}}{n}\bigg|{\mathcal C}_n\bigg)\leq E\bigg(\frac {g_{k}}{k}\bigg|{\mathcal C}_1\bigg) +\frac{3kC}{n} \mbox{ almost everywhere}\quad \mbox{for }0<k<n. \end{align*} $$

It follows that

$$ \begin{align*}\limsup_{n\to \infty}E\bigg(\frac {g_{n}}{n}\bigg|{\mathcal C}_n\bigg) \leq E\bigg(\frac {g_{k}}{k}\bigg|{\mathcal C}_1\bigg) \mbox{ almost everywhere}\quad\mbox{for each}\ k, \end{align*} $$

so by the dominated convergence theorem,

$$ \begin{align*} \limsup_{n\to \infty}E\bigg(\frac {g_{n}}{n}\bigg|{\mathcal C}_n\bigg) \leq \lim_{k\to \infty} E\bigg(\frac {g_{k}}{k}\bigg|{\mathcal C}_1\bigg)= E(g|{\mathcal C}_1)=g \mbox{ almost everywhere}. \end{align*} $$

Combining it with (2.13) yields the desired result $\lim _{n\to \infty }E( ({g_{n}}/{n})|{\mathcal C}_n)=g$ almost everywhere.

3 Some auxiliary results

In this section we give two auxiliary results (Propositions 3.1 and 3.4) which are needed in the proof of Theorem 1.1.

Proposition 3.1. Let $(X,\sigma )$ be a one-sided subshift over a finite alphabet ${\mathcal A}$ and $\mathcal {G}=\{g_n\}_{n=1}^\infty $ a subadditive potential on $\mathcal A^{\Bbb N}$ . Then

(3.1) $$ \begin{align} P(X,\sigma, \mathcal G)=\lim_{n\to \infty}\frac{1}{n} P(X^{(n)},\sigma^n, g_n)=\inf_{n\geq 1}\frac{1}{n} P(X^{(n)},\sigma^n, g_n), \end{align} $$

where $X^{(n)}$ is defined as in (2.7), and $P(X^{(n)},\sigma ^n, g_n)$ denotes the classical topological pressure of $g_n$ over the full shift space $(X^{(n)},\sigma ^n)$ .

Remark 3.2. Instead of (3.1), it was proved in [2, Proposition 2.2] that

$$ \begin{align*}P(X,\sigma, \mathcal G)=\lim_{n\to \infty}\frac{1}{n} P(X,\sigma^n, g_n)\end{align*} $$

under a more general setting. We remark that the proof of (3.1) is more subtle.

To prove Proposition 3.1, we need the following lemma.

Lemma 3.3. [10, Lemma 2.3]

Under the assumptions of Proposition 3.1, suppose that $\{\nu _n\}_{n=1}^\infty $ is a sequence in ${\mathcal M}(\mathcal A^{\Bbb N})$ , where ${\mathcal M}({\mathcal A}^{\Bbb N})$ denotes the space of all Borel probability measures on $\mathcal A^{\Bbb N}$ with the weak* topology. We form the new sequence $\{\mu _n\}_{n=1}^\infty $ by $\mu _n=(1/n) \sum _{i=0}^{n-1}\nu _n\circ \sigma ^{-i}$ . Assume that $\mu _{n_i}$ converges to $\mu $ in ${\mathcal M}({\mathcal A}^{\Bbb N})$ for some subsequence $\{n_i\}$ of natural numbers. Then $\mu \in {\mathcal M}({\mathcal A}^{\Bbb N}, \sigma )$ and, moreover,

$$ \begin{align*} \limsup_{i\to\infty}\frac{1}{n_i}\int g_{n_i}(x)\,d\nu_{n_i}(x)\leq \mathcal G_*(\mu):=\lim_{n\to\infty}\frac{1}{n}\int g_n\,d\mu. \end{align*} $$

Proof of Proposition 3.1

We first prove that, for each $n\in {\Bbb N}$ ,

(3.2) $$ \begin{align} P(X, \sigma, \mathcal G)\leq \frac{1}{n}P(X^{(n)},\sigma^n, g_n). \end{align} $$

To see this, fix $n\in {\Bbb N}$ and let $\mu $ be an equilibrium measure for the potential ${\mathcal G}$ . Then

$$ \begin{align*} P(X, \sigma, \mathcal G)&=h_\mu(\sigma)+\mathcal G_*(\mu)\\ &\leq h_\mu(\sigma)+\frac{1}{n}\int g_n\, d\mu\quad\mbox{(by (2.5))} \nonumber \\&=\frac{1}{n}\bigg( h_\mu(\sigma^n)+\int g_n\, d\mu\bigg)\\ &\leq \frac{1}{n}P(X^{(n)},\sigma^n, g_n), \end{align*} $$

where in the last inequality, we use the fact that $\mu \in \mathcal M(X^{(n)}, \sigma ^n)$ and the classical variational principle for the topological pressure of additive potentials. This proves (3.2).

In what follows we prove that

(3.3) $$ \begin{align} P(X, \sigma, \mathcal G)\geq \limsup_{n\to \infty}\frac{1}{n}P(X^{(n)},\sigma^n, g_n). \end{align} $$

Clearly (3.2) and (3.3) imply (3.1). To prove (3.3), by the classical variational principle we can take a subsequence $\{n_i\}$ of natural numbers and $\nu _{n_i}\in \mathcal M(X^{(n_i)}, \sigma ^{n_i})$ such that

(3.4) $$ \begin{align} \limsup_{n\to \infty}\frac{1}{n}P(X^{(n)},\sigma^n, g_n)=\lim_{i\to \infty}\frac{1}{n_i} \bigg(h_{\nu_{n_i}}(\sigma^{n_i})+\int g_{n_i}\, d\nu_{n_i}\bigg). \end{align} $$

Set $\mu ^{(i)}=({1}/{n_i})\sum _{k=0}^{n_i-1}\nu _{n_i}\circ \sigma ^{-k}$ for each i. Taking a subsequence if necessary, we may assume that $\mu ^{(i)}$ converges to an element $\mu \in \mathcal M(\mathcal A^{\Bbb N})$ in the weak* topology. By Lemma 3.3, $\mu \in {\mathcal M}({\mathcal A}^{\Bbb N}, \sigma )$ and, moreover,

(3.5) $$ \begin{align} \limsup_{i\to\infty}\frac{1}{n_i}\int g_{n_i}(x)\,d\nu_{n_i}(x)\leq \mathcal G_*(\mu). \end{align} $$

Next, we show that $\mu $ is supported on X. For this we adopt some arguments from the proof of [30, Theorem 1.1]. Notice that, for each i, $\mu ^{(i)}$ is $\sigma $ -invariant supported on

$$ \begin{align*} \bigcup_{k=0}^{n_i-1}\sigma^k X^{(n_i)}=\bigcup_{k=1}^{n_i}\sigma^{n_i-k} X^{(n_i)}. \end{align*} $$

Hence, $\mu $ is supported on

$$ \begin{align*} \bigcap_{N=1}^\infty \overline{ \bigcup_{i=N}^\infty \bigcup_{k=1}^{n_i} \sigma^{n_i-k} X^{(n_i)}}. \end{align*} $$

If x is in this set, then, for each $N\geq 1$ , there exist integers $i(N)\geq N$ and $k(N)\in [1, n_{i(N)}]$ for which $d(x, \sigma ^{n_{i(N)}-k(N)}X^{(n_{i(N)})})<1/N$ (i.e., $d(x,z)<1/N$ for some $z\in \sigma ^{n_{i(N)}-k(N)}X^{(n_{i(N)})}$ ), hence

(3.6) $$ \begin{align} d(x, X)\leq d(x,z)+d(z, X)< \frac{1}{N}+ \sup_{y\in \sigma^{n_{i(N)}-k(N)}X^{(n_{i(N)})} } d(y, X) \leq \frac{1}{N}+2^{-k(N)} \end{align} $$

and

(3.7) $$ \begin{align} d(\sigma^{k(N)}x, X)&\leq d(\sigma^{k(N)}x,\sigma^{k(N)}z)+d(\sigma^{k(N)}z, X)\nonumber\\ &\leq 2^{k(N)}d(x,z)+d(\sigma^{k(N)}z, X) \nonumber \\&<\frac{2^{k(N)}}{N}+\sup_{y\in \sigma^{n_{i(N)}-k(N)}X^{(n_{i(N)})} } d(\sigma^{k(N)}y, X)\nonumber\\ &\leq \frac{2^{k(N)}}{N}+2^{-n_{i(N)}}\quad \mbox{(since}\ \sigma^{k(N)}y\in \sigma^{n_{i(N)}}X^{(n_{i(N)})}=X^{(n_{i(N)})}\mbox{)}. \end{align} $$

If the values $k(N)$ are unbounded as $N\to \infty $ , then (3.6) yields $x\in X$ , while if they are bounded then some value of k recurs infinitely often as $k(N)$ , which implies that $\sigma ^kx\in X$ by (3.7). Thus $\mu $ is supported on

$$ \begin{align*} \bigcup_{k=0}^\infty\sigma^{-k}X. \end{align*} $$

Since $\sigma X\subset X$ , the set $(\sigma ^{-1} X)\backslash X $ is wandering under $\sigma ^{-1}$ (i.e., its preimages under powers of $\sigma $ are disjoint), so it must have zero $\mu $ -measure. Consequently, $\mu \in \mathcal M(X,\sigma )$ .

Notice that $h_{\mu ^{(i)}}(\sigma )=({1}/{n_i})h_{\nu _{n_i}}(\sigma ^{n_i})$ (see Lemma 2.4). By the upper semi-continuity of the entropy map,

$$ \begin{align*} h_\mu(\sigma)\geq \limsup_{i\to \infty}h_{\mu^{(i)}}(\sigma)=\limsup_{i\to \infty}\frac{1}{n_i}h_{\nu_{n_i}}(\sigma^{n_i}), \end{align*} $$

which, together with (3.5), yields that

$$ \begin{align*} h_\mu(\sigma)+\mathcal G_*(\mu)&\geq \limsup_{i\to \infty}\frac{1}{n_i} \bigg(h_{\nu_{n_i}}(\sigma^{n_i})+\int g_{n_i} \, d\nu_{n_i}\bigg)\\ &=\limsup_{n\to \infty}\frac{1}{n}P(X^{(n)},\sigma^n, g_n). \end{align*} $$

Applying Theorem 2.2, we obtain (3.3). This completes the proof of the proposition.

Next, we present another auxiliary result.

Proposition 3.4. Let $(X,\sigma )$ be a one-sided subshift over a finite alphabet ${\mathcal A}$ and $g, h\in C(X)$ . Assume, in addition, that $h(x)<0$ for all $x\in X$ . Let

$$ \begin{align*}r_0=\sup_{x\in X}\exp(h(x)).\end{align*} $$

Set, for $0<r<r_0$ ,

$$ \begin{align*} \mathcal A_r:=\big\{i_1\ldots i_n\in X^*: \sup_{x\in [i_1\ldots i_n]\cap X}\exp(S_nh(x))<r\leq \sup_{y\in [i_1\ldots i_{n-1}]\cap X}\exp(S_{n-1}h(y))\big\}, \end{align*} $$

where $X^*$ is the collection of finite words allowed in X and $S_nh(x):=\sum _{k=0}^{n-1} h(\sigma ^kx)$ . Then

(3.8) $$ \begin{align} \lim_{r\to 0}\frac{\log ( \sum_{I\in \mathcal A_r}\sup_{x\in [I]\cap X} \exp(S_{|I|}g(x)) )}{\log r}=-t, \end{align} $$

where t is the unique real number such that $P(X,\sigma , g+th)=0$ , and $|I|$ stands for the length of I.

To prove the above result, we need the following lemma.

Lemma 3.5. Let $(X,\sigma )$ be a one-sided subshift over a finite alphabet ${\mathcal A}$ and $f\in C(X)$ . Then

$$ \begin{align*} \lim_{n\to \infty} \frac{1}{n} \sup\{|S_nf(x)-S_nf(y)|: x_i=y_i \mbox{ for all } 1\leq i\leq n\}=0. \end{align*} $$

Proof. The result is well known. For the reader’s convenience, we include a proof.

Define, for $n\in {\Bbb N}$ ,

$$ \begin{align*}\mbox{var}_nf=\sup\{|f(x)-f(y)|: x_i=y_i \mbox{ for all } 1\leq i\leq n\}.\end{align*} $$

Since f is uniformly continuous, $\mbox {var}_nf\to 0$ as $n\to \infty $ . It follows that

$$ \begin{align*}\lim_{n\to \infty} \frac{1}{n}\sum_{i=1}^n \mbox{var}_if=0.\end{align*} $$

This concludes the result of the lemma since

$$ \begin{align*}\sup\{|S_nf(x)-S_nf(y)|: x_i=y_i \mbox{ for all } 1\leq i\leq n\}\end{align*} $$

is bounded above by $\sum _{i=1}^n \mbox {var}_if$ .

Proof of Proposition 3.4

Set

$$ \begin{align*} \Theta_r=\sum_{I\in \mathcal A_r}\sup_{x\in [I]\cap X} \exp(S_{|I|}g(x)),\quad r\in (0, r_0). \end{align*} $$

Let $\epsilon>0$ . It is enough to show that

(3.9) $$ \begin{align} r^{-t+\epsilon}\leq \Theta_r\leq r^{-t-\epsilon} \end{align} $$

for sufficiently small r.

To this end, set, for $0<r<r_0$ ,

$$ \begin{align*} m(r)=\min\{|I|: I\in \mathcal A_r\}, \quad M(r)=\max\{|I|: I\in \mathcal A_r\}. \end{align*} $$

From the definition of $\mathcal A_r$ and the negativity of h, it follows that there exist two positive constants $a, b$ such that

(3.10) $$ \begin{align} a\log(1/r)\leq m(r) \leq M(r)\leq b \log (1/r)\quad \text{for all } r\in (0, r_0). \end{align} $$

Define

$$ \begin{align*} \Gamma_r=\sum_{I\in \mathcal A_r}\sup_{x\in [I]\cap X} \exp(S_{|I|}(g+th)(x)),\quad r\in (0, r_0). \end{align*} $$

By Lemma 3.5 and (3.10), it is readily checked that

(3.11) $$ \begin{align} r^{t+\epsilon/2} \Theta_r\leq \Gamma_r\leq r^{t-\epsilon/2} \Theta_r \quad \mbox{for sufficiently small }r, \end{align} $$

Hence, to prove (3.9), it suffices to prove that $r^{\epsilon /2}\leq \Gamma _r\leq r^{-\epsilon /2}$ for small r.

We first prove $\Gamma _r>r^{\epsilon /2}$ when r is small. Suppose to the contrary that this is not true. Then by (3.10) we can find some $r\in (0, r_0)$ and $\unicode{x3bb}>0$ such that $Z(r, \unicode{x3bb} )<1$ , where

(3.12) $$ \begin{align} Z(r, \unicode{x3bb}):=\sum_{I\in \mathcal A_r}\exp(\unicode{x3bb} |I|) \sup_{x\in [I]\cap X} \exp(S_{|I|}(g+th)(x)). \end{align} $$

Observe that $\{[I]: I\in \mathcal A_r\}$ is a cover of X. From [7] it follows that $P(X, \sigma , g+th)\leq -\unicode{x3bb} $ , contradicting the fact that $P(X, \sigma , g+th)=0$ . Hence, we have $\Gamma _r>r^{\epsilon /2}$ when r is sufficiently small.

Next, we prove the inequality $\Gamma _r\leq r^{-\epsilon /2}$ for small r. To do this, fix $\unicode{x3bb} \in (0, \epsilon /(2b))$ , where b is the constant in (3.10). We claim that there exists $0<r_1<r_0$ such that

(3.13) $$ \begin{align} Z(r, -\unicode{x3bb})< 1\quad \mbox{for all } r\in (0, r_1), \end{align} $$

where Z is defined as in (3.12). Since $\unicode{x3bb} \in (0, \epsilon /(2b))$ , it follows from (3.10) that, for any $I\in \mathcal A_r$ ,

$$ \begin{align*}\exp(-\unicode{x3bb}|I|)\geq \exp(-\unicode{x3bb} b\log(1/r))=r^{b\unicode{x3bb}}\geq r^{\epsilon/2}.\end{align*} $$

Hence, (3.13) implies that $\Gamma _r\leq r^{-\epsilon /2}$ for $0<r<r_1$ .

Now it remains to prove (3.13). Since $P(X, \sigma , g+th)=0$ , by definition we have

$$ \begin{align*} \lim_{n\to \infty} \frac{1}{n}\log \bigg(\sum_{I\in X^*: \; |I|=n} \sup_{x\in [I]\cap X} \exp(S_{|I|}(g+th)(x))\bigg)=0. \end{align*} $$

Hence, there exists a large N such that $e^{-\unicode{x3bb} N/2}<1-e^{-\unicode{x3bb} /2}$ and, for any $n>N$ ,

$$ \begin{align*} \gamma_n:=\sum_{I\in X^*: \; |I|=n} \exp(-\unicode{x3bb} |I|) \sup_{x\in [I]\cap X} \exp(S_{|I|}(g+th)(x))\leq \exp(-\unicode{x3bb} n/2). \end{align*} $$

Take a small $r_1\in (0, r_0)$ so that $m(r)\geq N$ for any $0<r<r_1$ . By the definition of $m(r)$ , for any $0<r<r_1$ we have $\mathcal A_r\subset \{I\in X^*:|I|\geq N\}$ and so

$$ \begin{align*} Z(r, -\unicode{x3bb})\leq \sum_{n=N}^\infty \gamma_n\leq \sum_{n=N}^\infty \exp(-\unicode{x3bb} n/2)=\frac{e^{-\unicode{x3bb} N/2}}{1-e^{-\unicode{x3bb}/2}}<1. \end{align*} $$

This proves (3.13).

4 The proof of Theorem 1.1

Recall that, for $T\in {\Bbb R}^{d\times d}$ , $\alpha _1(T)\geq \cdots \geq \alpha _d(T)$ are the singular values of T, and $\phi ^s(T)$ ( $s\geq 0$ ) is defined as in (2.9). We begin with an elementary but important lemma.

Lemma 4.1. Let $E\subset U\subset {\Bbb R}^d$ , where E is compact and U is open. Let $k\in \{0,1,\ldots , d-1\}$ . Then, for any non-degenerate $C^1$ map $f: U\to {\Bbb R}^d$ , there exists $r_0>0$ such that, for any $y\in E$ , $z\in B(y, r_0)$ and $0<r<r_0$ , the set $f(B(z, r))$ can be covered by

$$ \begin{align*} (4d)^d\cdot\frac{\phi^k(D_yf)}{(\alpha_{k+1}(D_yf))^k} \end{align*} $$

balls of radius $\alpha _{k+1}(D_yf)r$ .

Proof. The result was implicitly proved in [40, Lemma 3] by using an idea of [12]. For the convenience of the reader, we provide a detailed proof.

Set

$$ \begin{align*} \gamma=\min_{y\in E}\alpha_d(D_yf). \end{align*} $$

Then

(4.1) $$ \begin{align} B(0, \gamma)\subset D_yf(B(0,1)) \quad\mbox{for all }y\in E. \end{align} $$

Since f is $C^1$ , non-degenerate on U and E is compact, it follows that $\gamma>0$ . Take $\epsilon =(2\sqrt {d}-1)/2$ . Then there exists a small $r_0>0$ such that, for $u, v, w\in V_{2r_0}(E):=\{x: d(x, E)<2r_0\}$ ,

(4.2) $$ \begin{align} |f(u)-f(v)-D_vf(u-v)|\leq \epsilon \gamma |u-v| \quad \mbox{ if }|u-v|\leq r_0, \end{align} $$

and

(4.3) $$ \begin{align} D_vf(B(0,1))\subset ((1+\epsilon)D_wf)(B(0,1))\quad \mbox{ if }|v-w|\leq r_0. \end{align} $$

Now let $y\in E$ and $z\in B(y, r_0)$ . For any $0<r<r_0$ and $x\in B(z, r)$ , taking $u=x$ and $v=z$ in (4.2) gives

$$ \begin{align*} f(x)-f(z)-D_zf(x-z)\in B(0, \epsilon \gamma r), \end{align*} $$

so by (4.3) and (4.1),

$$ \begin{align*} f(x)-f(z) &\in D_zf(B(0,r))+B(0, \epsilon\gamma r)\\ &\quad\subset ((1+\epsilon) D_yf)(B(0,r))+B(0, \epsilon \gamma r) \quad (\mbox{by (4.3)})\\ &\quad\subset D_yf(B(0,(1+\epsilon)r)+ B(0,\epsilon r)) \;\quad (\mbox{by (4.1)})\\ &\quad\subset D_yf(B(0,(1+2\epsilon)r))\\ &\quad = D_yf(B(0,2\sqrt{d}r)), \end{align*} $$

where $A+A':=\{u+v: u\in A, \; v\in A'\}$ . Therefore,

$$ \begin{align*} f(B(z, r))\subset f(z)+ D_yf(B(0,2\sqrt{d}r)). \end{align*} $$

That is, $f(B(z, r))$ is contained in an ellipsoid which has principle axes of lengths $4\sqrt {d}\alpha _i(D_yf)r$ , $i=1,\ldots , d$ . Hence, $f(B(z, r))$ is contained in a rectangular parallelepiped of side lengths $2\sqrt {d}\alpha _i(D_yf)r$ , $i=1,\ldots , d$ . Now we can divide such a parallelepiped into at most

$$ \begin{align*} \bigg(\prod_{i=1}^{k+1}\frac{4d\alpha_i(D_yf)}{\alpha_{k+1}(D_yf)}\bigg)\cdot (4d)^{d-k-1} \leq (4d)^d \cdot\frac{\phi^k(D_yf)}{(\alpha_{k+1}(D_yf))^k} \end{align*} $$

cubes of side $({2}/{\sqrt {d}})\cdot \alpha _{k+1}(D_yf)r$ . Therefore, this parallelepiped (and $f(B(z, r))$ as well) can be covered by

$$ \begin{align*}(4d)^d \cdot\frac{\phi^k(D_yf)}{(\alpha_{k+1}(D_yf))^k}\end{align*} $$

balls of radius $\alpha _{k+1}(D_yf)r$ .

In the remainder of this section let $\{f_i\}_{i=1}^\ell $ be a $C^1$ IFS on ${\Bbb R}^d$ with attractor K. Let $(\Sigma ,\sigma )$ be the one-sided full shift over the alphabet $\{1,\ldots , \ell \}$ and $\Pi : \Sigma \to K$ the canonical coding map associated with the IFS (cf. (1.3)). As a consequence of Lemma 4.1, we obtain the following proposition.

Proposition 4.2. Let $k\in \{0,1,\ldots , d-1\}$ . Set $C=(4d)^d$ . Then there exists $C_1>0$ such that, for $\mathbf {i}=(i_p)_{p=1}^\infty \in \Sigma $ and $n\in {\Bbb N}$ , the set $f_{\mathbf {i}|n}(K)$ can be covered by $C_1\prod _{p=0}^{n-1}G(\sigma ^p \mathbf {i})$ balls of radius $\prod _{p=0}^{n-1}H(\sigma ^p \mathbf {i})$ , where

(4.4) $$ \begin{align} G(\mathbf{i}):=\frac{C\phi^k(D_{\Pi \sigma \mathbf{i}}f_{i_1})}{\alpha_{k+1}(D_{\Pi \sigma \mathbf{i}}f_{i_1})^k},\quad H(\mathbf{i}):=\alpha_{k+1}(D_{\Pi \sigma \mathbf{i}}f_{i_1}). \end{align} $$

Proof. Since $\{f_i\}_{i=1}^\ell $ is a $C^1$ IFS, there exists an open set $U\supset K$ such that each $f_i$ extends to a $C^1$ diffeomorphism $f_i: U\to f_i(U)$ . Applying Lemma 4.1 to the mappings $f_i$ , we see that there exists $r_0>0$ such that, for any $y\in K$ , $z\in B(y, r_0)$ , $0<r<r_0$ and $i\in \{1,\ldots , \ell \}$ , the set $f_i(B(z, r))$ can be covered by

$$ \begin{align*}\theta(y,i):=\frac{C\phi^k(D_{y}f_{i})}{\alpha_{k+1}(D_{y}f_{i})^k}\end{align*} $$

balls of radius $\alpha _{k+1}(D_{y}f_{i})r$ .

Since $f_1,\ldots , f_\ell $ are contracting on U, there exists $\gamma \in (0,1)$ such that

$$ \begin{align*} |f_i(x)-f_i(y)|\leq \gamma|x-y| \quad\mbox{for all }x, y\in U,\; i\in \{1,\ldots, \ell\}. \end{align*} $$

This implies that $\alpha _{1}(D_yf_i)\leq \gamma $ for any $y\in K$ and $i\in \{1,\ldots , \ell \}$ . Take a large integer $n_0$ such that

$$ \begin{align*}\gamma^{n_0}\max\{1,{\textrm{diam}}(K)\}<r_0/2.\end{align*}$$

Clearly there exists a large number $C_1$ so that the conclusion of the proposition holds for any positive integer $n\leq n_0$ and $\mathbf {i}\in \Sigma $ , that is, the set $f_{\mathbf {i}|n}(K)$ can be covered by $C_1\prod _{p=0}^{n-1}G(\sigma ^p \mathbf {i})$ balls of radius $\prod _{p=0}^{n-1}H(\sigma ^p \mathbf {i})$ . Below we show by induction that this holds for all $n\in {\Bbb N}$ and $\mathbf {i}\in \Sigma $ .

Suppose, for some $m\geq n_0$ , that the conclusion of the proposition holds for any positive integer $n\leq m$ and $\mathbf {i}\in \Sigma $ . Then, for ${\textbf {i}}\in \Sigma $ , $f|_{(\sigma {\textbf {i}})|m}(K)$ can be covered by $C_1\prod _{p=0}^{m-1}G(\sigma ^{p+1} \mathbf {i})$ balls of radius $\prod _{p=0}^{m-1}H(\sigma ^{p+1} \mathbf {i})$ . Let $B_1,\ldots B_N$ denote these balls. We may assume that $B_j\cap f|_{(\sigma {\textbf {i}})|m}(K)\neq \emptyset $ for each j. Since

$$ \begin{align*}\prod_{p=0}^{m-1}H(\sigma^{p+1} \mathbf{i})\leq \gamma^m\leq \gamma^{n_0}<r_0/2\end{align*} $$

and

$$ \begin{align*}d(\Pi\sigma\mathbf{i}, B_j\cap f_{(\sigma {\textbf{i}})|m}(K))\leq {\textrm{diam}}(f_{(\sigma {\textbf{i}})|m}(K))\leq \gamma^{n_0}{\textrm{diam}}(K)<r_0/2,\end{align*} $$

the center of $B_j$ is in $B(\Pi \sigma {\textbf {i}}, r_0)$ . Therefore, $f_{i_1}(B_j)$ can be covered by $\theta (\Pi \sigma \mathbf {i}, i_1)=G(\bf {i})$ balls of radius

$$ \begin{align*}H({\textbf{i}})\cdot \prod_{p=0}^{m-1}H(\sigma^{p+1} \mathbf{i})=\prod_{p=0}^{m}H(\sigma^{p} \mathbf{i}).\end{align*} $$

Since $f_{\mathbf {i}|{(m+1)}}(K)\subset \bigcup _{j=1}^N f_{i_1}(B_j)$ , it follows that $f_{\mathbf {i}|{(m+1)}}(K)$ can be covered by

$$ \begin{align*}G({\textbf{i}})N\leq G({\textbf{i}}) \cdot C_1\prod_{p=0}^{m-1}G(\sigma^{p+1} {\textbf{i}})=C_1\prod_{p=0}^{m}G(\sigma^{p} {\textbf{i}})\end{align*} $$

balls of radius $\prod _{p=0}^{m}H(\sigma ^{p} \mathbf {i})$ . Thus the proposition also holds for $n=m+1$ and all ${\textbf {i}}\in \Sigma $ , as desired.

Next, we provide an upper bound on the upper box-counting dimension of the attractor K of the IFS $\{f_i\}_{i=1}^\ell $ .

Proposition 4.3. Let $k\in \{0, 1,\ldots , d-1\}$ . Let $G, H:\Sigma \to {\Bbb R}$ be defined as in (4.4). Let t be the unique real number so that

$$ \begin{align*}P(\Sigma,\sigma,(\log G)+ t(\log H))=0.\end{align*} $$

Then $ \overline {\dim }_BK\leq t. $

Proof. Write $g=\log G$ and $h=\log H$ for short. Define

$$ \begin{align*} r_{\min}=\min_{x\in \Sigma}\alpha_{k+1}(D_{\Pi\sigma x}f_{x_1}),\quad r_{\max}=\max_{x\in \Sigma}\alpha_{k+1}(D_{\Pi\sigma x}f_{x_1}). \end{align*} $$

Then $0<r_{\min } \leq r_{\max }<1$ . For $0<r<r_{\min }$ , define

(4.5) $$ \begin{align} \mathcal A_r=\big\{i_1\ldots i_n\in \Sigma^*: \sup_{x\in [i_1\ldots i_n]}S_nh(x)<\log r\leq \sup_{y\in [i_1\ldots i_{n-1}]}S_{n-1}h(y)\big\}; \end{align} $$

clearly $\{[I]: I\in \mathcal A_r\}$ is a partition of $\Sigma $ . By Proposition 4.2, there exists a constant $C_1>0$ such that, for each $0<r<r_{\min }$ , every $I\in \mathcal A_r$ and $x\in [I]$ , $f_I(K)$ can be covered by

$$ \begin{align*} C_1\exp(S_{|I|}g(x)) \leq C_1\exp\big(\sup_{y\in [I]}S_{|I|}g(y)\big) \end{align*} $$

balls of radius

$$ \begin{align*}\exp(S_{|I|}h(x))\leq \exp\big(\sup_{y\in [I]}S_{|I|}h(y)\big)<r.\end{align*} $$

It follows that K can be covered by

$$ \begin{align*} C_1\sum_{I\in \mathcal A_r}\exp\big(\sup_{y\in [I]}S_{|I|}g(y)\big) \end{align*} $$

balls of radius r. Hence, by Proposition 3.4,

$$ \begin{align*} \overline{\dim}_BK\leq \limsup_{r\to 0}\frac{\log\big(\sum_{I\in \mathcal A_r}\exp\big(\sup_{y\in [I]}S_{|I|}g(y)\big)\big)}{\log (1/r)}= t. \end{align*} $$

This completes the proof of the proposition.

As an application of Proposition 4.3, we may estimate the upper box-counting dimension of the projections of a class of $\sigma $ -invariant sets under the coding map.

Proposition 4.4. Let X be a compact subset of $\Sigma $ satisfying $\sigma X\subset X$ , and $k\in \{0,\ldots , d-1\}$ . Then, for each $n\in {\Bbb N}$ ,

$$ \begin{align*} \overline{\dim}_B\Pi(X^{(n)})\leq t_n, \end{align*} $$

where $X^{(n)}$ is defined as in (2.7), $t_n$ is the unique number for which

$$ \begin{align*}P(X^{(n)}, \sigma^n, (\log G_n)+ t_n(\log H_n))=0,\end{align*} $$

and $G_n, H_n$ are continuous functions on $\Sigma $ defined by

(4.6) $$ \begin{align} G_n(y):=\frac{C\phi^k(D_{\Pi \sigma^n y}f_{y|n})}{\alpha_{k+1}(D_{\Pi \sigma^n y}f_{y|n})^k},\quad H_n(y):=\alpha_{k+1}(D_{\Pi \sigma^n y}f_{y|n}), \end{align} $$

with $C=(4d)^d$ .

Proof. The result is obtained by applying Proposition 4.3 to the IFS $\{f_{I}: I\in X^*_n\}$ instead of $\{f_i\}_{i=1}^\ell $ , where $X_n^*$ stands for the collection of words of length n allowed in X.

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1

Write $s=\dim _SX$ . We may assume that $s<d$ ; otherwise we have nothing left to prove. Set $k=[s]$ , that is, k is the largest integer less than or equal to s. Let $\mathcal U=\{u_n\}_{n=1}^\infty $ be the subadditive potential on $\Sigma $ defined by

$$ \begin{align*}u_n(x)=\log \phi^s(D_{\Pi\sigma^n x} f_{x|n}).\end{align*} $$

Then $P(X, \sigma , {\mathcal U})=0$ by the definition of $\dim _SX$ . Hence, by Proposition 3.1,

$$ \begin{align*} \lim_{n\to \infty}\frac{1}{n}P(X^{(n)}, \sigma^n, u_n)=\inf_{n\geq 1}\frac{1}{n}P(X^{(n)}, \sigma^n, u_n)=0. \end{align*} $$

It follows that, for each $\epsilon>0$ , there exists $N_\epsilon>0$ such that

(4.7) $$ \begin{align} 0\leq P(X^{(n)}, \sigma^n, u_n)\leq n\epsilon \quad\mbox{for }\; n>N_\epsilon. \end{align} $$

For $n\in {\Bbb N}$ , let $s_n$ be the unique real number such that $P(X^{(n)}, \sigma ^n, v_n)=0$ , where $v_n$ is a continuous function on $\Sigma $ defined by

$$ \begin{align*} v_n(x)&= \log G_n(x)+ s_n \log H_n(x)\\ &=\log ( C\phi^k(D_{\Pi\sigma^nx}f_{x|n})\alpha_{k+1}(D_{\Pi\sigma^nx}f_{x|n})^{s_n-k}), \end{align*} $$

where $G_n, H_n$ are defined in (4.6) and $C=(4d)^d$ . By Proposition 4.4,

$$ \begin{align*}\overline{\dim}_B\Pi(X)\leq \overline{\dim}_B\Pi(X^{(n)})\leq s_n.\end{align*} $$

If $s\geq s_n$ for some n, then $\overline {\dim }_B\Pi (X)\leq s_n\leq s$ and we are done. In what follows, we assume that $s<s_n$ for each n. Then, for each $x\in \Sigma $ ,

$$ \begin{align*} u_n(x)-v_n(x)&= -\log C +(s-s_n)\log \alpha_{k+1}(D_{\Pi\sigma^nx}f_{x|n})\\ &\geq -\log C +(s-s_n)\log \alpha_1(D_{\Pi\sigma^nx}f_{x|n})\\ &\geq -\log C+ n(s-s_n) \log \theta, \end{align*} $$

where

$$ \begin{align*} \theta:=\max_{y\in \Sigma} \alpha_1(D_{\Pi\sigma y}f_{y_1})<1. \end{align*} $$

Hence,

$$ \begin{align*} P(X^{(n)}, \sigma^n, u_n)&=P(X^{(n)}, \sigma^n, u_n)-P(X^{(n)}, \sigma^n, v_n)\\ & \geq \inf_{x\in \Sigma} (u_n(x)-v_n(x))\\ &\geq -\log C+ n(s-s_n) \log \theta, \end{align*} $$

where in the second inequality, we used [39, Theorem 9.7(iv)]. Combining this with (4.7) yields that, for $n\geq N_\epsilon $ ,

$$ \begin{align*} n\epsilon\geq -\log C + n(s-s_n) \log \theta, \end{align*} $$

so

$$ \begin{align*} s\geq s_n + \frac{\epsilon+ n^{-1} \log C}{\log \theta}\geq \overline{\dim}_B\Pi(X)+\frac{\epsilon+ n^{-1} \log C}{\log \theta}. \end{align*} $$

Letting $n\to \infty $ and then $\epsilon \to 0$ , we obtain $s\geq \overline {\dim }_B\Pi (X)$ , as desired.

5 The proof of Theorem 1.2

Let $\Pi :\Sigma \to {\Bbb R}^d$ be the coding map associated with a $C^1$ IFS $\{f_i\}_{i=1}^\ell $ on ${\Bbb R}^d$ (cf. (1.3)). For $E\subset {\Bbb R}^d$ and $\delta>0$ , let $N_\delta (E)$ denote the smallest integer N for which E can be covered by N closed balls of radius $\delta $ . For $T\in {\Bbb R}^{d\times d}$ , let $\alpha _1(T)\geq \cdots \geq \alpha _d(T)$ denote the singular values of T, and let $\phi ^s(T)$ be the singular value function defined as in (2.9).

The following geometric counting lemma plays an important role in the proof of Theorem 1.2. It is of independent interest as well.

Lemma 5.1. Let m be an ergodic $\sigma $ -invariant Borel probability measure on $\Sigma $ . Set

(5.1) $$ \begin{align} \unicode{x3bb}_i:=\lim_{n\to \infty} \frac{1}{n}\int \log( \alpha_i(D_{\Pi\sigma^nx}f_{x|n}))\,dm(x),\quad i=1,\ldots, d. \end{align} $$

Let $k\in \{0,\ldots , d-1\}$ . Write $u:=\exp (\unicode{x3bb} _{k+1})$ . Then, for m-a.e. $x\in \Sigma $ ,

(5.2) $$ \begin{align} \limsup_{n\to \infty} \frac{1}{n}\log N_{u^n}(\Pi([x|n]))\leq (\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k)-k\unicode{x3bb}_{k+1}. \end{align} $$

Proof. It is known (see, for example, [1, Theorem 3.3.3]) that, for m-a.e. x,

$$ \begin{align*} \lim_{p\to \infty}\frac{1}{p} \log (\alpha_i (D_{\Pi \sigma^p x}f_{x|p}))=\unicode{x3bb}_i,\quad i=1,\ldots, d, \end{align*} $$

and

$$ \begin{align*} \lim_{p\to \infty}\frac{1}{p} \log (\phi^i (D_{\Pi \sigma^p x}f_{x|p}))=\unicode{x3bb}_1+\cdots+\unicode{x3bb}_i,\quad i=1,\ldots, d. \end{align*} $$

For $i\in \{1,\ldots , d\}$ , $x\in \Sigma $ and $p\in {\Bbb N}$ , set

$$ \begin{align*} \begin{split} w_p^{(i)}(x)&= \log \phi^i(D_{\Pi \sigma^p x}f_{x|p}), \\ v_p^{(i)}(x)&= \log \alpha_i(D_{\Pi \sigma^p x}f_{x|p}). \end{split} \end{align*} $$

Then, by the definition of $\phi ^i$ ,

(5.3) $$ \begin{align} w_p^{(i)}=v_p^{(1)}+\cdots+v_p^{(i)}. \end{align} $$

Since $\phi ^i$ is submultiplicative on ${\Bbb R}^{d\times d}$ (cf. [13, Lemma 2.1]), $\{w^{(i)}_p\}_{p=1}^\infty $ is a subadditive potential satisfying

$$ \begin{align*} |w_p^{(i)}(x)|\leq pC,\quad p\in {\Bbb N}, x\in \Sigma, \end{align*} $$

for some constant $C>0$ . Set ${\mathcal C}_p:=\{B\in {\mathcal B}(\Sigma ): \sigma ^{-p}B=B\ \mathrm {almost\ everywhere}\}$ . Then, by Lemma 2.8, for m-a.e. x,

(5.4) $$ \begin{align} \lim_{p\to \infty}E\bigg(\frac{w^{(i)}_p}{p} \bigg| {\mathcal C}_p \bigg)(x) =\lim_{p\to \infty} \frac{1}{p}w^{(i)}_p(x)=\unicode{x3bb}_1+\cdots+\unicode{x3bb}_i, \quad i=1,\ldots, d, \end{align} $$

and so by (5.3),

(5.5) $$ \begin{align} \lim_{p\to \infty}E\bigg(\frac{v^{(i)}_p}{p} \bigg| {\mathcal C}_p \bigg)(x) =\unicode{x3bb}_i, \quad i=1,\ldots, d. \end{align} $$

Let $p\in {\Bbb N}$ . Applying Proposition 4.2 to the IFS $\{f_I: I\in \mathcal A^p\}$ , we see that there exists a positive number $C_1(p)$ such that, for any $x\in \Sigma $ and $n\in {\Bbb N}$ , the set $\Pi ([x|np])=f_{x|np}(K)$ can be covered by $C_1(p)\prod _{i=0}^{n-1}G_p(\sigma ^{pi} x)$ balls of radius $\prod _{i=0}^{n-1}H_p(\sigma ^{ip} x)$ , where

(5.6) $$ \begin{align} G_p(x):=\frac{C\phi^k(D_{\Pi \sigma^px }f_{x|p})}{\alpha_{k+1}(D_{\Pi \sigma^p x}f_{x|p})^k},\quad H_p(x):=\alpha_{k+1}(D_{\Pi \sigma^px}f_{x|p}), \end{align} $$

with $C=(4d)^d$ . By Birkhoff’s ergodic theorem, for m-a.e. $x\in \Sigma $ ,

(5.7) $$ \begin{align} \lim_{n\to \infty}\frac{1}{np}\log \bigg(\!C_1(p)\prod_{i=0}^{n-1}G_p(\sigma^{pi} x)\!\bigg)&=\frac{\log C}{p}+E\bigg(\frac{w^{(k)}_p}{p} \bigg| {\mathcal C}_p \!\bigg)(x) -kE\bigg(\frac{v^{(k+1)}_p}{p} \bigg| {\mathcal C}_p \!\bigg)(x) \end{align} $$

and

(5.8) $$ \begin{align} \lim_{n\to \infty}\frac{1}{np}\log \bigg( \prod_{i=0}^{n-1}H_p(\sigma^{ip} x)\bigg)=E\bigg(\frac{v^{(k+1)}_p}{p} \bigg| {\mathcal C}_p \bigg)(x). \end{align} $$

Let $\epsilon>0$ . By (5.4), (5.5), (5.7), (5.8), for m-a.e. x there exists a positive integer $p_0(x)$ such that, for any $p\geq p_0(x)$ ,

(5.9) $$ \begin{align} \prod_{i=0}^{n-1}H_p(\sigma^{ip} x)\leq (u+\epsilon)^{np}\quad \mbox{for large enough }n, \end{align} $$

and

(5.10) $$ \begin{align} \frac{1}{np}\log \bigg( C_1(p)\prod_{i=0}^{n-1}G_p(\sigma^{pi} x)\bigg) \leq \unicode{x3bb}_1+\cdots +\unicode{x3bb}_k-k\unicode{x3bb}_{k+1}+\epsilon \quad \mbox{for large enough }n. \end{align} $$

Fix such an x and let $p\geq p_0(x)$ . By (5.9),

$$ \begin{align*} N_{(u+\epsilon)^{np}}(\Pi([x|pn]))\leq C_1(p)\prod_{i=0}^{n-1}G_p(\sigma^{pi} x) \quad \mbox{for large enough }n. \end{align*} $$

Notice that there exists a constant $C_2=C_2(d)>0$ such that a ball of radius $(u+\epsilon )^{np}$ in ${\Bbb R}^d$ can be covered by $C_2 (1+\epsilon /u)^{dnp}$ balls of radius $u^{np}$ . It follows that, for large enough n,

$$ \begin{align*} N_{u^{np}}(\Pi([x|pn]))&\leq C_2 (1+\epsilon/u)^{dnp} N_{(u+\epsilon)^{np}}(\Pi([x|pn]))\\ &\leq C_1(p) C_2 (1+\epsilon/u)^{dnp} \prod_{i=0}^{n-1}G_p(\sigma^{pi} x). \end{align*} $$

Hence, by (5.10),

$$ \begin{align*} \limsup_{n\to \infty}\frac{1}{n}\log N_{u^{n}}(\Pi([x|n]))&=\limsup_{n\to \infty} \frac{1}{np}\log N_{u^{np}}(\Pi([x|pn]))\\ &\leq d\log(1+\epsilon/u)+ \limsup_{n\to \infty} \frac{1}{np}\log \bigg(\prod_{i=0}^{n-1}G_p(\sigma^{pi} x)\bigg)\\ &\leq d\log(1+\epsilon/u)+\epsilon+\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k-k\unicode{x3bb}_{k+1}, \end{align*} $$

where the first equality follows from the fact that, for $pn\leq m<p(n+1)$ ,

$$ \begin{align*} N_{u^{m}}(\Pi([x|m]))\leq N_{u^{m}}(\Pi([x|pn]))&\leq 4^d (u^{pn-m})^d N_{u^{np}}(\Pi([x|pn]))\\ &\leq 4^d u^{-pd} N_{u^{np}}(\Pi([x|pn])), \end{align*} $$

using the fact that, for $R>r>0$ , a ball of radius R in ${\Bbb R}^d$ can be covered by $(4R/r)^d$ balls of radius r. Letting $\epsilon \to 0$ yields the desired inequality (5.2).

The following result is also needed in the proof of Theorem 1.2.

Lemma 5.2. Let m be a Borel probability measure on $\Sigma $ . Let $\rho , \epsilon \in (0,1)$ . Then, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma $ ,

(5.11) $$ \begin{align} m\circ\Pi^{-1} (B(\Pi x, 2\rho^n))\geq (1-\epsilon)^n\frac{m([x_1\ldots x_n])}{N_{\rho^n}(\Pi([x_1\ldots x_n]))} \quad \mbox{for large enough }n. \end{align} $$

Proof. The formulation and the proof of the above lemma are adapted from an argument given by Jordan [25]. A similar idea was also employed in the proof of [36, Theorem 2.2].

For $n\in {\Bbb N}$ , let $\Lambda _n$ denote the set of the points $x=(x_n)_{n=1}^\infty \in \Sigma $ such that

$$ \begin{align*} m\circ\Pi^{-1} (B(\Pi x, 2\rho^n))< (1-\epsilon)^n\frac{m([x_1\ldots x_n])}{N_{\rho^n}(\Pi([x_1\ldots x_n]))}. \end{align*} $$

To prove that (5.11) holds almost everywhere, by the Borel–Cantelli lemma it suffices to show that

(5.12) $$ \begin{align} \sum_{n=1}^\infty m(\Lambda_n)<\infty. \end{align} $$

For this purpose, let us estimate $m(\Lambda _n)$ . Fix $n\in {\Bbb N}$ and $I\in \mathcal A^n$ . Notice that $\Pi ([I])$ can be covered by $N_{\rho ^n}(\Pi ([I]))$ balls of radius $\rho ^n$ . As a consequence, there exists $L\leq N_{\rho ^n}(\Pi ([I]))$ such that $\Pi (\Lambda _n\cap [I])$ can be covered by L balls of radius $\rho ^n$ , say, $B_1,\ldots , B_L$ . We may assume that $\Pi (\Lambda _n\cap [I])\cap B_i\neq \emptyset $ for each $1\leq i\leq L$ . Hence, for each i, we may pick $x^{(i)}\in \Lambda _n \cap [I]$ such that $\Pi x^{(i)} \in B_i$ . Clearly $B_i\subset B(\Pi x^{(i)}, 2 \rho ^n)$ . Since $x^{(i)}\in \Lambda _n \cap [I]$ , by the definition of $\Lambda _n$ we obtain

$$ \begin{align*}m\circ\Pi^{-1} (B(\Pi x^{(i)}, 2\rho^n))< (1-\epsilon)^n\frac{m([I])}{N_{\rho^n}(\Pi([I]))}.\end{align*} $$

It follows that

$$ \begin{align*} m(\Lambda_n\cap [I])&\leq m\circ \Pi^{-1}(\Pi(\Lambda_n\cap [I]))\\ &\leq m\circ \Pi^{-1}\bigg(\bigcup_{i=1}^LB_i\bigg)\\ &\leq m\circ \Pi^{-1}\bigg(\bigcup_{i=1}^LB(\Pi x^{(i)}, 2\rho^n)\bigg)\\ &\leq L (1-\epsilon)^n\frac{m([I])}{N_{\rho^n}(\Pi([I]))}\\ &\leq (1-\epsilon)^nm([I]). \end{align*} $$

Summing over $I\in \mathcal A^n$ yields that $m(\Lambda _n)\leq (1-\epsilon )^n$ , which implies (5.12).

Remark 5.3. Lemma 5.2 remains valid when the coding map $\Pi :\Sigma \to {\Bbb R}^d$ is replaced by any Borel measurable map from $\Sigma $ to ${\Bbb R}^d$ .

We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2

We may assume that $s:=\dim _{L}m<d$ ; otherwise there is nothing left to prove. Set $k=[s]$ . Let $\unicode{x3bb} _i$ , $i=1,\ldots , d$ , be defined as in (5.1). Then, by Definition 2.6,

$$ \begin{align*} h_m(\sigma)+\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k+(s-k)\unicode{x3bb}_{k+1}=0. \end{align*} $$

Let $u=\exp (\unicode{x3bb} _{k+1})$ and $\epsilon \in (0,1)$ . Applying Lemma 5.2 yields that, for m-a.e. $x=(x_n)_{n=1}^\infty \in ~\Sigma $ ,

(5.13) $$ \begin{align} m\circ\Pi^{-1} (B(\Pi x, 2u^n))\geq (1-\epsilon)^n\frac{m([x_1\ldots x_n])}{N_{u^n}(\Pi([x_1\ldots x_n]))} \quad \mbox{for large enough }n. \end{align} $$

It follows that, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma $ ,

$$ \begin{align*} &\overline{d}(m\circ \Pi^{-1}, \Pi x)\\ &\quad=\limsup_{n\to \infty} \frac{\log(m\circ\Pi^{-1} (B(\Pi x, 2u^n)))}{n\log u} \\ &\quad\leq \frac{\log(1-\epsilon)}{\log u}+\limsup_{n\to \infty}\bigg(\frac{\log m([x_1\ldots x_n])}{n \log u} -\frac{\log N_{u^n}(\Pi([x_1\ldots x_n]))}{n\log u}\bigg)\\ &\quad\leq \frac{\log(1-\epsilon)-h_m(\sigma)-(\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k)+k\unicode{x3bb}_{k+1}}{\log u}\\ &\quad= \frac{\log(1-\epsilon)+s\unicode{x3bb}_{k+1}}{\unicode{x3bb}_{k+1}}, \end{align*} $$

where in the third inequality, we used the Shannon–McMillan–Breiman theorem (cf. [39, p. 93]) and Lemma 5.1 (keeping in mind that $\log u=\unicode{x3bb} _{k+1}<0$ ). Letting $\epsilon \to 0$ yields the desired result.

6 Upper bound for the box-counting dimension of $C^1$ -repellers and the Lyapunov dimensions of ergodic invariant measures

Throughout this section let ${\pmb M}$ be a smooth Riemannian manifold of dimension d and $\psi : {\pmb M}\to {\pmb M}$ a $C^1$ -map. Let $\Lambda $ be a compact subset of ${\pmb M}$ such that $\psi (\Lambda )=\Lambda $ , and assume that $\Lambda $ is a repeller of $\psi $ (cf. §1). Below we first introduce the definitions of singularity and Lyapunov dimensions for the case of repellers, which are sightly different from that for the case of IFSs.

Definition 6.1. The singularity dimension of $\Lambda $ with respect to $\psi $ , written as $\dim _{S^*}\Lambda $ , is the unique real value s for which

$$ \begin{align*} P(\Lambda, \psi, {\mathcal G}^ s)=0, \end{align*} $$

where ${\mathcal G}^s=\{g^s_n\}_{n=1}^\infty $ is the subadditive potential on $\Lambda $ defined by

(6.1) $$ \begin{align} g^s_n(z)=\log \phi^s((D_z \psi^n)^{-1}),\quad z\in \Lambda. \end{align} $$

Definition 6.2. For an ergodic $\psi $ -invariant measure $\mu $ supported on $\Lambda $ , the Lyapunov dimension of $\mu $ with respect to $\psi $ , written as $\dim _{L^*}\mu $ , is the unique real value s for which

$$ \begin{align*} h_\mu(\psi)+ \mathcal G^s_*(\mu)=0, \end{align*} $$

where ${\mathcal G}^s=\{g^s_n\}_{n=1}^\infty $ is defined as in (6.1) and $\mathcal G^s_*(\mu )=\lim _{n\to \infty }({1}/{n}) \int g^s_n\,d\mu $ .

Before proving Theorems 1.3–1.4, we recall some definitions and necessary facts about $C^1$ repellers.

A finite closed cover $\{R_1,\ldots ,R_{\ell }\}$ of $\Lambda $ is called a Markov partition of $\Lambda $ with respect to $\psi $ if:

1. (i) $\overline {\mbox {int}R_i}=R_i$ for each $i=1,\ldots ,\ell $ ;

2. (ii) $\mbox {int}R_i \cap \mbox {int}R_j=\emptyset $ for $i\neq j$ ; and

3. (iii) each $\psi (R_i)$ is the union of a subfamily of $\{R_j\}_{j=1}^{\ell }$ .

It is well known that any repeller of an expanding map has Markov partitions of arbitrary small diameter (see [37, p. 146]). Let $\{R_1,\ldots , R_\ell \}$ be a Markov partition of $\Lambda $ with respect to $\psi $ . It is known that this dynamical system induces a subshift space of finite type $(\Sigma _A,\sigma )$ over the alphabet $\{1,\ldots , \ell \}$ , where $A=(a_{ij})$ is the transfer matrix of the Markov partition, namely, $a_{ij}=1$ if $\mbox {int}R_i\cap \psi ^{-1}(\mbox {int}R_j)\neq \emptyset $ and $a_{ij}=0$ otherwise [37], and

$$ \begin{align*} \Sigma_A=\{(i_n)_{n=1}^\infty\in \{1,\ldots, \ell\}^{\Bbb N}: a_{i_ni_{n+1}}=1 \mbox{ for all }n\geq 1\}. \end{align*} $$

This gives the coding map $\Pi : \Sigma _A\rightarrow \Lambda $ such that

(6.2) $$ \begin{align} \Pi({\textbf{i}})=\bigcap_{n\geq 1}\psi^{-(n-1)}(R_{i_n}) \quad \text{for all } \ {\textbf{i}}=(i_n)_{n=1}^\infty\in \Sigma_A, \end{align} $$

and the following diagram commutes:

(6.3)

(Keep in mind that throughout this section, $\Pi $ denotes the coding map for the repeller $\Lambda $ and no longer for the coding map for an IFS as used in the previous sections.)

The coding map $\Pi $ is a Hölder continuous surjection. Moreover, there is a positive integer q such that $\Pi ^{-1}(z)$ has at most q elements for each $z\in \Lambda $ (see [37, p. 147]).

For $n\geq 1$ , define

$$ \begin{align*}\Sigma_{A,n}:=\{i_1\ldots i_n\in \{1,\ldots, \ell\}^n: a_{i_ki_{k+1}}=1 \mbox{ for } 1\leq k\leq n-1\}.\end{align*} $$

For any word $I=i_1\ldots i_n\in \Sigma _{A, n}$ , the set $\bigcap _{k=1}^{n}\psi ^{-(k-1)}(R_{i_{k}})$ is called a basic set and is denoted by $R_{I}$ .

The proof of Theorem 1.3 is similar to that of Theorem 1.1. We begin with the following lemma, which is a slight variant of Lemma 4.1.

Lemma 6.3. Let $E\subset U\subset {\pmb M}$ , where E is compact and U is open. Let $k\in \{0,1,\ldots , d-1\}$ . Then, for any non-degenerate $C^1$ map $f: U\to {\pmb M}$ , there exists $r_0>0$ so that, for any $y\in E$ , $z\in B(y, r_0)$ and $0<r<r_0$ , the set $f(B(z, r))$ can be covered by

$$ \begin{align*} C_{\pmb M}\cdot\frac{\phi^k(D_yf)}{(\alpha_{k+1}(D_yf))^k} \end{align*} $$

balls of radius $\alpha _{k+1}(D_yf)r$ , where $C_{\pmb M}$ is a positive constant depending on ${\pmb M}$ .

Proof. This can be done by routinely modifying the proof of Lemma 4.1 and using similar arguments to the proof of [40, Corollary 1].

Let $\delta>0$ be small enough so that $\psi : B(z, \delta )\to \psi (B(z, \delta ))$ is a diffeomorphism for each z in the $\delta $ -neighborhood of $\Lambda $ . Suppose that $\{R_1,\ldots , R_\ell \}$ is a Markov partition of $\Lambda $ with diameter less than $\delta $ .

The following result is an analogue of Proposition 4.2.

Proposition 6.4. Let $k\in \{0,1,\ldots , d-1\}$ . Set $C_{\pmb M}$ be the constant in Lemma 6.3. Then there exists $C_1>0$ such that, for all $\mathbf {i}=(i_p)_{p=1}^\infty \in \Sigma _A$ and $n\in {\Bbb N}$ , the basic set $R_{\mathbf {i}|n}$ can be covered by $C_1\prod _{p=0}^{n-1}G(\sigma ^p \mathbf {i})$ balls of radius $\prod _{p=0}^{n-1}H(\sigma ^p \mathbf {i})$ , where

(6.4) $$ \begin{align} G(\mathbf{i}):=\frac{C_{\pmb M}\phi^k((D_{\Pi \mathbf{i}}\psi)^{-1})}{\alpha_{k+1}((D_{\Pi \mathbf{i}}\psi)^{-1})^k},\quad H(\mathbf{i}):=\alpha_{k+1}((D_{\Pi \mathbf{i}} \psi)^{-1}). \end{align} $$

Remark 6.5. The definitions of the functions G and H in the above proposition are slightly different from that in Proposition 4.2.

Proof of Proposition 6.4

The proof is adapted from that of Proposition 4.2. For the reader’s convenience, we provide the full details.

First, we construct a local inverse $f_{i,j}$ of $\psi $ for each pair $(i,j)$ with $ij\in \Sigma _{A,2}$ . To do so, notice that $\psi (R_{ij})=R_j$ for each $ij\in \Sigma _{A,2}$ . Since $\psi $ is a diffeomorphism restricted on a small neighborhood of $R_i$ , we can find open sets $\widetilde {R}_{ij}$ and $\widetilde {R_j}$ such that $\widetilde {R}_{ij} \supset R_{ij}$ , $\widetilde {R_j}\supset R_j$ , $\psi (\widetilde {R}_{ij})=\widetilde {R_j}$ and $\psi : \widetilde {R}_{ij}\to \widetilde {R_j}$ is diffeomorphic. Then we take $f_{i,j}: \widetilde {R_j}\to \widetilde {R}_{ij}$ to be the inverse of $\psi : \widetilde {R}_{ij}\to \widetilde {R_j}$ , and the construction is done.

For any ${\textbf {i}}=(i_n)_{n=1}^\infty \in \Sigma _A$ , we see that $\Pi \sigma {\textbf {i}}\in R_{i_2}\subset \widetilde {R_{i_2}}$ and $(\psi \circ f_{i_1, i_2})|_{\widetilde {R_{i_2}}}$ is the identity restricted on $\widetilde {R_{i_2}}$ . Since $\psi (\Pi {\textbf {i}})=\Pi \sigma {\textbf {i}}$ , it follows that $f_{i_1, i_2}(\Pi \sigma {\textbf {i}})=\Pi {\textbf {i}}$ . Differentiating $\psi \circ f_{i_1, i_2}$ at $\Pi \sigma {\textbf {i}}$ and applying the chain rule, we get

$$ \begin{align*} (D_{\Pi {\textbf{i}}} \psi)(D_{\Pi\sigma {\textbf{i}}}f_{i_1, i_2})=\mbox{Identity}, \end{align*} $$

so

(6.5) $$ \begin{align} D_{\Pi\sigma {\textbf{i}}}f_{i_1, i_2}=(D_{\Pi {\textbf{i}}} \psi)^{-1}. \end{align} $$

According to Lemma 6.3, there exists $r_0>0$ such that, for each $ij\in \Sigma _{A,2}$ , $y\in R_j$ , $z\in B(y, r_0)$ and $0<r<r_0$ , the set $f_{i,j}(B(z, r))$ can be covered by

$$ \begin{align*} C_{\pmb M}\cdot\frac{\phi^k(D_yf_{i,j})}{(\alpha_{k+1}(D_yf_{i,j}))^k} \end{align*} $$

balls of radius $\alpha _{k+1}(D_yf_{i,j})r$ .

Since $\psi $ is expanding on $\Lambda $ , there exists $\gamma \in (0,1)$ such that $\sup _{{\textbf {i}}\in \Sigma _A}\alpha _1 ((D_{\Pi \mathbf {i}} \psi )^{-1})<\gamma $ . Then $\sup _{{\textbf {i}}\in \Sigma _A}H({\textbf {i}})<\gamma $ and

(6.6) $$ \begin{align} \mbox{diam}(R_{{\textbf{i}}|(n+1)})\leq \gamma \mbox{diam}(R_{(\sigma{\textbf{i}})|n}) \end{align} $$

for all ${\textbf {i}}\in \Sigma _A$ and $n\in {\Bbb N}$ . Take a large integer $n_0$ such that

(6.7) $$ \begin{align} \gamma^{n_0-1} \max\{1, \mbox{diam}(\Lambda)\}<r_0/2. \end{align} $$

By (6.6)–(6.7), $\mbox {diam}(R_{{\textbf {i}}|n})<r_0/2$ for all ${\textbf {i}}\in \Sigma _A$ and $n\geq n_0$ .

Clearly there exists a large number $C_1$ so that the conclusion of the proposition holds for any positive integer $n\leq n_0$ and $\mathbf {i}\in \Sigma _A$ , that is, the set $R_{\mathbf {i}|n}$ can be covered by $C_1\prod _{p=0}^{n-1}G(\sigma ^p \mathbf {i})$ balls of radius $\prod _{p=0}^{n-1}H(\sigma ^p \mathbf {i})$ . Below we show by induction that this holds for all $n\in {\Bbb N}$ and $\mathbf {i}\in \Sigma _A$ .

Suppose, for some $m\geq n_0$ , that the conclusion of the proposition holds for any positive integer $n\leq m$ and $\mathbf {i}\in \Sigma _A$ . Then, for given ${\textbf {i}}=(i_n)_{n=1}^\infty \in \Sigma _A$ , $R_{(\sigma {\textbf {i}})|m}$ can be covered by $C_1\prod _{p=0}^{m-1}G(\sigma ^{p+1} \mathbf {i})$ balls of radius $\prod _{p=0}^{m-1}H(\sigma ^{p+1} \mathbf {i})$ . Let $B_1,\ldots , B_N$ denote these balls. We may assume that $B_j\cap R_{(\sigma {\textbf {i}})|m}\neq \emptyset $ for each j. Since

$$ \begin{align*}\prod_{p=0}^{m-1}H(\sigma^{p+1} \mathbf{i})\leq \gamma^m\leq \gamma^{n_0}<r_0/2\end{align*} $$

and

$$ \begin{align*}d(\Pi\sigma\mathbf{i}, B_j\cap R_{(\sigma {\textbf{i}})|m})\leq {\textrm{diam}}(R_{(\sigma {\textbf{i}})|m})<r_0/2,\end{align*} $$

so the center of $B_j$ is in $B(\Pi \sigma {\textbf {i}}, r_0)$ . Therefore, by Lemma 6.3 and (6.5), $f_{i_1, i_2}(B_j)$ can be covered by

$$ \begin{align*} C_{\pmb M}\cdot\frac{\phi^k(D_{\Pi \sigma {\textbf{i}}}f_{i_1, i_2})}{(\alpha_{k+1}(D_{\Pi \sigma {\textbf{i}}}f_{i_1, i_2}))^k}=G({\textbf{i}}) \end{align*} $$

balls of radius

$$ \begin{align*}\alpha_{k+1}(D_{\Pi\sigma{\textbf{i}}}f_{i_1,i_2})\cdot\prod_{p=0}^{m-1}H(\sigma^{p+1} \mathbf{i})= H({\textbf{i}})\cdot \prod_{p=0}^{m-1}H(\sigma^{p+1} \mathbf{i})=\prod_{p=0}^{m}H(\sigma^{p} \mathbf{i}).\end{align*} $$

Since $\psi (R_{\mathbf {i}|{(m+1)}})\subset R_{(\sigma {\textbf {i}})|m}$ , it follows that

$$ \begin{align*}R_{\mathbf {i}|{(m+1)}}\subset f_{i_1, i_2} (R_{(\sigma {\textbf{i}})|m})\subset \bigcup_{j=1}^N f_{i_1, i_2}(B_j),\end{align*} $$

hence $R_{\mathbf {i}|{(m+1)}}$ can be covered by

$$ \begin{align*}G({\textbf{i}})N\leq G({\textbf{i}}) \cdot C_1\prod_{p=0}^{m-1}G(\sigma^{p+1} {\textbf{i}})=C_1\prod_{p=0}^{m}G(\sigma^{p} {\textbf{i}})\end{align*} $$

balls of radius $\prod _{p=0}^{m}H(\sigma ^{p} \mathbf {i})$ . Thus the proposition also holds for $n=m+1$ and all ${\textbf {i}}\in \Sigma _A$ , as desired.

Proposition 6.6. Let $k\in \{0, 1,\ldots , d-1\}$ . Let $G, H:\Sigma _A\to {\Bbb R}$ be defined as in (6.4). Let t be the unique real number so that

$$ \begin{align*}P(\Sigma_A,\sigma,(\log G)+ t(\log H))=0.\end{align*} $$

Then $ \overline {\dim }_B\Lambda \leq t. $

Proof. Here we use similar arguments to that in the proof of Proposition 4.3. Write $g=\log G$ and $h=\log H$ . Define

$$ \begin{align*} r_{\min}=\min_{{\textbf{i}}\in \Sigma_A}h({\textbf{i}}),\quad r_{\max}=\max_{{\textbf{i}}\in \Sigma_A}h({\textbf{i}}). \end{align*} $$

Then $0<r_{\min }\leq r_{\max }<1$ . For $0<r<r_{\min }$ , define

(6.8) $$ \begin{align} \kern-8pt\mathcal A_r=\big\{i_1\ldots i_n\in \Sigma^*_A: \sup_{x\in [i_1\ldots i_n]\cap \Sigma_A}S_nh(x)<\log r\leq \sup_{y\in [i_1\ldots i_{n-1}]\cap \Sigma_A}S_{n-1}h(y)\big\}, \end{align} $$

where $\Sigma _A^*$ denotes the set of all finite words allowed in $\Sigma _A$ . Clearly $\{[I]: I\in \mathcal A_r\}$ is a partition of $\Sigma _A$ . By Proposition 6.4, there exists a constant $C_1>0$ such that, for each $0<r<r_{\min }$ , every $I\in \mathcal A_r$ and $x\in [I]$ , $R_I$ can be covered by

$$ \begin{align*} C_1\exp(S_{|I|}g(x)) \leq C_1\exp\big(\sup_{y\in [I]}S_{|I|}g(y)\big) \end{align*} $$

balls of radius

$$ \begin{align*}\exp(S_{|I|}h(x))\leq \exp\big(\sup_{y\in [I]}S_{|I|}h(y)\big)<r.\end{align*} $$

It follows that $\Lambda $ can be covered by

$$ \begin{align*} C_1\sum_{I\in \mathcal A_r}\exp\big(\sup_{y\in [I]}S_{|I|}g(y)\big) \end{align*} $$

balls of radius r. Hence, by Proposition 3.4,

$$ \begin{align*} \overline{\dim}_B\Lambda\leq \limsup_{r\to 0}\frac{\sum_{I\in \mathcal A_r}\exp\big(\sup_{y\in [I]}S_{|I|}g(y)\big)}{\log (1/r)}= t. \end{align*} $$

This completes the proof of the proposition.

For $n\in {\Bbb N}$ , applying Proposition 6.6 to the mapping $\psi ^n$ instead of $\psi $ , we obtain the following result.

Proposition 6.7. Let $k\in \{0,\ldots , d-1\}$ . Then, for each $n\in {\Bbb N}$ ,

$$ \begin{align*} \overline{\dim}_B\Lambda\leq t_n, \end{align*} $$

where $t_n$ is the unique number for which $P(\Sigma _A, \sigma ^n, (\log G_n)+ t_n(\log H_n))=0$ , and $G_n, H_n$ are continuous functions on $\Sigma _A$ defined by

(6.9) $$ \begin{align} G_n(y):=\frac{C\phi^k((D_{\Pi y}\psi^n)^{-1})}{\alpha_{k+1}((D_{\Pi y}\psi^n)^{-1}))^k},\quad H_n(y):=\alpha_{k+1}((D_{\Pi y}\psi^n)^{-1})), \end{align} $$

with $C=C_{\pmb M}$ being the constant in Lemma 6.3.

We are now ready to prove Theorem 1.3.

Proof of Theorem 1.3

We follow the proof of Theorem 1.1 with slight modifications. Write $s=\dim _{S^*}(\Lambda )$ . We may assume that $s<d$ ; otherwise we have nothing left to prove. Set $k=[s]$ . Let $\mathcal G^s=\{g_n^s\}_{n=1}^\infty $ be the subadditive potential on $\Lambda $ defined by

$$ \begin{align*}g_n^s(z)=\log \phi^s((D_{z} \psi^n)^{-1}).\end{align*} $$

Then $P(\Lambda , \psi , {\mathcal G}^s)=0$ by the definition of $\dim _{S^*}(\Lambda )$ . Let $\widehat {\mathcal G}^s:=\{\widehat {g}_n^s\}_{n=1}^\infty $ , where $\widehat {g}_n^s\in C(\Sigma _A)$ is defined by

$$ \begin{align*} \widehat{g}^s_n({\textbf{i}})=g_n^s(\Pi{\textbf{i}})=\log \phi^s((D_{\Pi{\textbf{i}}} \psi^n)^{-1}),\quad {\textbf{i}}\in \Sigma_A. \end{align*} $$

Clearly, $\widehat {\mathcal G}^s$ is a subadditive potential on $\Sigma _A$ and

$$ \begin{align*} (\widehat{\mathcal G}^s)_*(m)=({\mathcal G}^s)_*(m\circ \Pi^{-1}), \quad m\in {\mathcal M}(\Sigma_A, \sigma). \end{align*} $$

Since the factor map $\Pi :\Sigma _A\to \Lambda $ is onto and finite-to-one, by Lemma 2.3, $m\to m\circ \Pi ^{-1}$ is a surjective map from ${\mathcal M}(\Sigma _A, \sigma )$ to $\mathcal M(\Lambda , \psi )$ and, moreover, $h_m(\sigma )=h_{m\circ \Pi ^{-1}}(\psi )$ for $m\in \mathcal M (\Sigma _A, \sigma )$ . By the variational principle for the subadditive pressure (see Theorem 2.2),

$$ \begin{align*} P(\Lambda, \psi, \mathcal{G}^{s}) &= \sup \{h_{\mu}(\psi)+(\mathcal{G}^{s})_{*}(\mu) : \mu \in \mathcal{M}(\Lambda, \psi)\}\\ &= \sup \{h_{m \circ \Pi^{-1}}(\psi)+(\mathcal{G}^{s})_{*}(m \circ \Pi^{-1}) : m \in \mathcal{M}(\Sigma_{A}, \sigma)\} \\ &= \sup \{h_{m}(\sigma)+(\widehat{\mathcal{G}}^{s})_{*}(m) : m \in \mathcal{M}(\Sigma_{A}, \sigma)\} \\ &= P(\Sigma_{A},\sigma,\widehat{\mathcal{G}}^{s}). \end{align*} $$

It follows that $P(\Sigma _{A},\sigma ,\widehat {\mathcal {G}}^{s})=0$ .

By [2],

$$ \begin{align*} \lim_{n\to \infty}\frac{1}{n}P(\Sigma_A, \sigma^n, \widehat{g}^s_n)=\inf_{n\geq 1}\frac{1}{n}P(\Sigma_A, \sigma^n, \widehat{g}^s_n)=P(\Sigma_A, \sigma, \widehat{\mathcal G}^s). \end{align*} $$

Since $P(\Sigma _{A},\sigma ,\widehat {\mathcal {G}}^{s})=0$ , the above equalities imply that, for each $\epsilon>0$ , there exists $N_\epsilon>0$ such that

(6.10) $$ \begin{align} 0\leq P(\Sigma_A, \sigma^n, \widehat{g}^s_n)\leq n\epsilon \quad\mbox{for }\; n>N_\epsilon. \end{align} $$

For $n\in {\Bbb N}$ , let $s_n$ be the unique real number such that $P(\Sigma _A, \sigma ^n, v_n^{s_n})=0$ , where $v_n^{s_n}$ is a continuous function on $\Sigma _A$ defined by

$$ \begin{align*} v_n^{s_n}(x)&= \log G_n(x)+ s_n \log H_n(x)\\ &=\log ( C\phi^k((D_{\Pi x}\psi^n)^{-1})\alpha_{k+1}((D_{\Pi x}\psi^n)^{-1})^{s_n-k}), \end{align*} $$

where $G_n, H_n$ are defined in (6.9) and $C=C_{\pmb M}$ . By Proposition 6.7,

$$ \begin{align*}\overline{\dim}_B\Lambda\leq s_n.\end{align*} $$

If $s\geq s_n$ for some n, then $\overline {\dim }_B\Lambda \leq s_n\leq s$ and we are done. In what follows, we assume that $s<s_n$ for each n. Then, for each $x\in \Sigma _A$ ,

$$ \begin{align*} \widehat{g}^s_n(x)-v_n^{s_n}(x)&= -\log C +(s-s_n)\log \alpha_{k+1}((D_{\Pi x}\psi^n)^{-1})\\ &\geq -\log C +(s-s_n)\log \alpha_1((D_{\Pi x}\psi^n)^{-1})\\ &\geq -\log C+ n(s-s_n) \log \theta, \end{align*} $$

where

$$ \begin{align*} \theta:=\max_{x\in \Sigma_A} \alpha_1((D_{\Pi x}\phi)^{-1})<1. \end{align*} $$

Hence,

$$ \begin{align*} P(\Sigma_A, \sigma^n,\widehat{g}^s_n)&=P(\Sigma_A, \sigma^n,\widehat{g}^s_n)-P(\Sigma_A, \sigma^n, v_n^{s_n})\\ &\geq \inf_{x\in \Sigma_A} ( \widehat{g}^s_n(x)-v_n^{s_n}(x))\\ &\geq -\log C+ n(s-s_n) \log \theta. \end{align*} $$

where in the second inequality, we used [39, Theorem 9.7(iv)]. Combining this with (6.10) yields that, for $n\geq N_\epsilon $ ,

$$ \begin{align*} n\epsilon\geq -\log C + n(s-s_n) \log \theta, \end{align*} $$

so

$$ \begin{align*} s\geq s_n + \frac{\epsilon+ n^{-1} \log C}{\log \theta}\geq \overline{\dim}_B\Lambda+\frac{\epsilon+ n^{-1} \log C}{\log \theta}. \end{align*} $$

Letting $n\to \infty $ and then $\epsilon \to 0$ , we obtain $s\geq \overline {\dim }_B\Lambda $ , as desired.

In the remainder of this section we prove Theorem 1.4. To this end, we need the following two lemmas, which are the analogues of Lemmas 5.1–5.2 for $C^1$ repellers.

Lemma 6.8. Let m be an ergodic $\sigma $ -invariant Borel probability measure on $\Sigma _A$ . Set

(6.11) $$ \begin{align} \unicode{x3bb}_i:=\lim_{n\to \infty} \frac{1}{n}\int \log( \alpha_i((D_{\Pi x}\psi^n)^{-1}))\,dm(x),\quad i=1,\ldots, d. \end{align} $$

Let $k\in \{0,\ldots , d-1\}$ . Write $u:=\exp (\unicode{x3bb} _{k+1})$ . Then, for m-a.e. $x\in \Sigma _A$ ,

(6.12) $$ \begin{align} \limsup_{n\to \infty} \frac{1}{n}\log N_{u^n}(R_{x|n})\leq (\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k)-k\unicode{x3bb}_{k+1}, \end{align} $$

where $N_\delta (E)$ is the smallest integer N for which E can be covered by N closed balls of radius $\delta $ .

Lemma 6.9. Let m be a Borel probability measure on $\Sigma _A$ . Let $\rho , \epsilon \in (0,1)$ . Then, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma _A$ ,

(6.13) $$ \begin{align} m\circ\Pi^{-1} (B(\Pi x, 2\rho^n))\geq (1-\epsilon)^n\frac{m([x_1\ldots x_n])}{N_{\rho^n}(R_{x_1\ldots x_n})} \quad \mbox{for large enough }n. \end{align} $$

The proofs of these two lemmas are essentially identical to those of Lemmas 5.1–5.2, so we omit them.

Proof of Theorem 1.4

Here we adapt the proof of Theorem 1.2. We may assume that $s:=\dim _{L^*}\mu <d$ ; otherwise there is nothing left to prove. Since $\Pi : \Sigma _A\to \Lambda $ is surjective and finite-to-one, by Lemma 2.3, there exists a $\sigma $ -invariant ergodic measure m on $\Sigma _A$ so that $m\circ \Pi ^{-1}=\mu $ and $h_m(\sigma )=h_\mu (\psi )$ .

Set $k=[s]$ . Let $\unicode{x3bb} _i$ , $i=1,\ldots , d$ , be defined as in (6.11). Then, by Definition 6.2,

$$ \begin{align*} h_m(\sigma)+\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k+(s-k)\unicode{x3bb}_{k+1}=0. \end{align*} $$

Let $u=\exp (\unicode{x3bb} _{k+1})$ and $\epsilon \in (0,1)$ . Applying Lemma 6.9 (in which we take $\rho =u$ ) yields that, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma _A$ ,

(6.14) $$ \begin{align} \mu (B(\Pi x, 2u^n))\geq (1-\epsilon)^n\frac{m([x_1\ldots x_n])}{N_{u^n}(R_{x_1\ldots x_n})} \quad \mbox{for large enough }n. \end{align} $$

It follows that, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma _A$ ,

$$ \begin{align*} \overline{d}(\mu, \Pi x) &=\limsup_{n\to \infty} \frac{\log(\mu (B(\Pi x, 2u^n)))}{n\log u} \\[6pt] &\leq \frac{\log(1-\epsilon)}{\log u}+\limsup_{n\to \infty}\bigg(\frac{\log m([x_1\ldots x_n])}{n \log u} -\frac{\log N_{u^n}(\Pi([x_1\ldots x_n]))}{n\log u}\bigg)\\[6pt] &\leq \frac{\log(1-\epsilon)-h_m(\sigma)-(\unicode{x3bb}_1+\cdots+\unicode{x3bb}_k)+k\unicode{x3bb}_{k+1}}{\log u}\\[3pt] &= \frac{\log(1-\epsilon)+s\unicode{x3bb}_{k+1}}{\unicode{x3bb}_{k+1}}, \end{align*} $$

where in the third inequality we used the Shannon–McMillan–Breiman theorem (cf. [39, p. 93]) and Lemma 6.8 (keeping in mind that $\log u=\unicode{x3bb} _{k+1}<0$ ). Letting $\epsilon \to 0$ yields the desired result.

Remark 6.10. There is an alternative way to prove Theorem 1.3 in the case when ${\pmb M}$ is an open set of ${\Bbb R}^d$ . We can construct a $C^1$ IFS $\{f_i\}_{i=1}^\ell $ on ${\Bbb R}^d$ so that $\Lambda $ is the projection of a shift-invariant set under the coding map associated with the IFS. Then we can use Theorem 1.1 to get an upper bound for $\overline {\dim }_B\Lambda $ . An additional effort is then required to justify that this upper bound is indeed equal to $\dim _{S^*}\Lambda $ . The details of this approach will be given in a forthcoming survey paper.