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 [Reference Bowen8, Reference Gatzouras and Peres23, Reference Ruelle38]. 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, [Reference Patzschke34]).
The study of dimension in the non-conformal cases has proved to be much more difficult. In his seminal paper [Reference Falconer13], 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 [Reference Bárány, Hochman and Rapaport3, Reference Hochman and Rapaport24] and the references therein. However, very little has been known in the higher-dimensional case.
In [Reference Falconer15, 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
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 [Reference Falconer15, Lemma 5.2]). Examples involving ‘triangular maps’ were constructed in [Reference Manning and Simon32] to show that the condition (1.1) is necessary for this bounded distortion property.
Using a quite different approach, Zhang [Reference Zhang40] 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 [Reference Ban, Cao and Hu2, 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 [Reference Zhang40, 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 [Reference Barreira4, Theorem 3], Barreira made a positive claim concerning this question, but his proof contains a crucial mistakeFootnote † , as found by Manning and Simon [Reference Manning and Simon32]. In a recent paper [Reference Cao, Pesin and Zhao9, 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 [Reference Feng and Simon22] 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
(cf. [Reference Falconer16]).
Let $(\Sigma ,\sigma )$ be the one-sided full shift over the alphabet $\{1,\ldots , \ell \}$ (cf. [Reference Bowen7]). Let $\Pi : \Sigma \to K$ denote the canonical coding map associated with the IFS $\{f_i\}_{i=1}^\ell $ . That is,
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. [Reference Falconer16]). 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,
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,
where $B(x,r)$ denotes the closed ball centered at x of radius r. Equivalently,
where $\dim _PF$ stands for the packing dimension of F (cf. [Reference Falconer16]). See, for example, [Reference Fan, Lau and Rao17] 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
where $m\circ \Pi ^{-1}$ stands for the push-forward of m by $\Pi $ .
The above theorem improves a result of Jordan and Pollicott [Reference Jordan and Pollicott26, Theorem 1] which states that
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 [Reference Jordan25] and Rossi [Reference Rossi36] 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:
-
(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 ${\pmb M}$ );
-
(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 [Reference Pesin35, §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
Theorem 1.4. Let $\mu $ be an ergodic $\psi $ -invariant measure supported on $\Lambda $ . Then
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 [Reference Falconer15] 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 [Reference Falconer15, 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 [Reference Douady and Oesterlé12] (see also [Reference Zhang40]) 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 [Reference Zhang40]. 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):
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 [Reference Jordan25, Reference Rossi36].
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
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
Following [Reference Cao, Feng and Huang10], below we define the topological pressure of $\mathcal {G}$ .
Definition 2.1. For given $n\in \mathbb {N}$ and $\varepsilon>0$ , we define
Then the topological pressure of $\mathcal {G}$ with respect to T is defined by
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, [Reference Walters39]).
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. [Reference Walters39]). Moreover, for $\mu \in \mathcal M(X, T)$ , by subadditivity we have
See, for example, [Reference Walters39, 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. [Reference Cao, Feng and Huang10, 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
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, [Reference Barreira5, Reference Falconer14, Reference Feng18, Reference Feng and Lau21, Reference Käenmäki28, Reference Mummert33] 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, [Reference Feng19, 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
for each $\mu \in {\mathcal M}(X_1, T_1)$ .
Proof. The first part of the result is the same as [Reference Mañé31, Ch. IV, Lemma 8.3]. The second part follows from the Abramov–Rokhlin formula (see [Reference Bogenschütz, Crauel, Krengel, Richter and Warstat6]).
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 $ :
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 $n\in {\Bbb N}$ ,
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
where $[\mathbf {i}]:=\{x\in \Sigma : x|n=\mathbf {i}\}$ for $\mathbf {i}\in \mathcal A^n$ ; see [Reference Cao, Feng and Huang10, 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 [Reference Falconer14] 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,
see, for example, [Reference Feng and Hu20, Lemma 4.6]. It follows that
This proves the claim.
By the affinity of the measure-theoretic entropy $h_{(\cdot )}(\sigma ^m)$ (see [Reference Walters39, Theorem 8.1]), we have
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 [Reference Falconer13], for $s\geq 0$ we define the singular value function $\phi ^s: {\Bbb R}^{d \times d}\to [0,\infty )$ as
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
where $\mathcal G^s=\{g_n^s\}_{n=1}^\infty $ is the subadditive potential on $\Sigma $ defined by
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
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
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 $d=2$ .
It follows from the definition of the singular value function $\phi ^s$ that, for an ergodic measure m, we have
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
-
(i) The concept of singularity dimension was first introduced by Falconer [Reference Falconer13, Reference Falconer15]; see also [Reference Käenmäki and Vilppolainen29]. It is also called affinity dimension when the IFS $\{f_i\}_{i=1}^\ell $ is affine, that is, each map $f_i$ is affine.
-
(ii) The definition of Lyapunov dimension of ergodic measures with respect to an IFS presented above was taken from [Reference Jordan and Pollicott26]. It is a generalization of that given in [Reference Jordan, Pollicott and Simon27] 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:
Suppose that there exists $C>0$ such that
Then
for m-a.e. x, where
$E(\cdot |\cdot )$ 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 $n\in {\Bbb N}$ ,
Since $\sum _{j=0}^{k-1}g_n(T^{jn}x)\geq g_{kn}(x)$ by subadditivity, it follows that
Since the sequence $\{g_n\}$ is subadditive and satisfies (2.12), by [Reference Käenmäki28, Lemma 2.2], for any $0< k< n$ ,
As a consequence,
Notice that, for each $f\in L^1$ and $n\in {\Bbb N}$ ,
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:
Now applying the identity (2.15) (with $f=g_k$ ) to (2.14) yields
It follows that
so by the dominated convergence theorem,
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
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 [Reference Ban, Cao and Hu2, Proposition 2.2] that
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. [Reference Cao, Feng and Huang10, 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,
Proof of Proposition 3.1
We first prove that, for each $n\in {\Bbb N}$ ,
To see this, fix $n\in {\Bbb N}$ and let $\mu $ be an equilibrium measure for the potential ${\mathcal G}$ . Then
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
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
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,
Next, we show that $\mu $ is supported on X. For this we adopt some arguments from the proof of [Reference Kenyon and Peres30, Theorem 1.1]. Notice that, for each i, $\mu ^{(i)}$ is $\sigma $ -invariant supported on
Hence, $\mu $ is supported on
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
and
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
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,
which, together with (3.5), yields that
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
Set, for $0<r<r_0$ ,
where $X^*$ is the collection of finite words allowed in X and $S_nh(x):=\sum _{k=0}^{n-1} h(\sigma ^kx)$ . Then
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
Proof. The result is well known. For the reader’s convenience, we include a proof.
Define, for $n\in {\Bbb N}$ ,
Since f is uniformly continuous, $\mbox {var}_nf\to 0$ as $n\to \infty $ . It follows that
This concludes the result of the lemma since
is bounded above by $\sum _{i=1}^n \mbox {var}_if$ .
Proof of Proposition 3.4
Set
Let $\epsilon>0$ . It is enough to show that
for sufficiently small r.
To this end, set, for $0<r<r_0$ ,
From the definition of $\mathcal A_r$ and the negativity of h, it follows that there exist two positive constants $a, b$ such that
Define
By Lemma 3.5 and (3.10), it is readily checked that
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
Observe that $\{[I]: I\in \mathcal A_r\}$ is a cover of X. From [Reference Bowen7] 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
where Z is defined as in (3.12). Since $\unicode{x3bb} \in (0, \epsilon /(2b))$ , it follows from (3.10) that, for any $I\in \mathcal A_r$ ,
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
Hence, there exists a large N such that $e^{-\unicode{x3bb} N/2}<1-e^{-\unicode{x3bb} /2}$ and, for any $n>N$ ,
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
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
balls of radius $\alpha _{k+1}(D_yf)r$ .
Proof. The result was implicitly proved in [Reference Zhang40, Lemma 3] by using an idea of [Reference Douady and Oesterlé12]. For the convenience of the reader, we provide a detailed proof.
Set
Then
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 $u, v, w\in V_{2r_0}(E):=\{x: d(x, E)<2r_0\}$ ,
and
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
where $A+A':=\{u+v: u\in A, \; v\in A'\}$ . Therefore,
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
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
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
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
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
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
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
and
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
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
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
Then $ \overline {\dim }_BK\leq t. $
Proof. Write $g=\log G$ and $h=\log H$ for short. Define
Then $0<r_{\min } \leq r_{\max }<1$ . For $0<r<r_{\min }$ , define
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
balls of radius
It follows that K can be covered by
balls of radius r. Hence, by Proposition 3.4,
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 $n\in {\Bbb N}$ ,
where $X^{(n)}$ is defined as in (2.7), $t_n$ is the unique number for which
and $G_n, H_n$ are continuous functions on $\Sigma $ defined by
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
Then $P(X, \sigma , {\mathcal U})=0$ by the definition of $\dim _SX$ . Hence, by Proposition 3.1,
It follows that, for each $\epsilon>0$ , there exists $N_\epsilon>0$ such that
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
where $G_n, H_n$ are defined in (4.6) and $C=(4d)^d$ . By Proposition 4.4,
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 $ ,
where
Hence,
where in the second inequality, we used [Reference Walters39, Theorem 9.7(iv)]. Combining this with (4.7) yields that, for $n\geq N_\epsilon $ ,
so
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
Let $k\in \{0,\ldots , d-1\}$ . Write $u:=\exp (\unicode{x3bb} _{k+1})$ . Then, for m-a.e. $x\in \Sigma $ ,
Proof. It is known (see, for example, [Reference Arnold1, Theorem 3.3.3]) that, for m-a.e. x,
and
For $i\in \{1,\ldots , d\}$ , $x\in \Sigma $ and $p\in {\Bbb N}$ , set
Then, by the definition of $\phi ^i$ ,
Since $\phi ^i$ is submultiplicative on ${\Bbb R}^{d\times d}$ (cf. [Reference Falconer13, Lemma 2.1]), $\{w^{(i)}_p\}_{p=1}^\infty $ is a subadditive potential satisfying
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,
and so by (5.3),
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
with $C=(4d)^d$ . By Birkhoff’s ergodic theorem, for m-a.e. $x\in \Sigma $ ,
and
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 $p\geq p_0(x)$ ,
and
Fix such an x and let $p\geq p_0(x)$ . By (5.9),
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,
Hence, by (5.10),
where the first equality follows from the fact that, for $pn\leq m<p(n+1)$ ,
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 $ ,
Proof. The formulation and the proof of the above lemma are adapted from an argument given by Jordan [Reference Jordan25]. A similar idea was also employed in the proof of [Reference Rossi36, 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
To prove that (5.11) holds almost everywhere, by the Borel–Cantelli lemma it suffices to show that
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
It follows that
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,
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 $ ,
It follows that, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma $ ,
where in the third inequality, we used the Shannon–McMillan–Breiman theorem (cf. [Reference Walters39, 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
where ${\mathcal G}^s=\{g^s_n\}_{n=1}^\infty $ is the subadditive potential on $\Lambda $ defined by
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
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:
-
(i) $\overline {\mbox {int}R_i}=R_i$ for each $i=1,\ldots ,\ell $ ;
-
(ii) $\mbox {int}R_i \cap \mbox {int}R_j=\emptyset $ for $i\neq j$ ; and
-
(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 [Reference Ruelle37, 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 [Reference Ruelle37], and
This gives the coding map $\Pi : \Sigma _A\rightarrow \Lambda $ such that
and the following diagram commutes:
(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 [Reference Ruelle37, p. 147]).
For $n\geq 1$ , define
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
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 [Reference Zhang40, 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
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
so
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
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
for all ${\textbf {i}}\in \Sigma _A$ and $n\in {\Bbb N}$ . Take a large integer $n_0$ such that
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
and
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
balls of radius
Since $\psi (R_{\mathbf {i}|{(m+1)}})\subset R_{(\sigma {\textbf {i}})|m}$ , it follows that
hence $R_{\mathbf {i}|{(m+1)}}$ can be covered by
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
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
Then $0<r_{\min }\leq r_{\max }<1$ . For $0<r<r_{\min }$ , define
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
balls of radius
It follows that $\Lambda $ can be covered by
balls of radius r. Hence, by Proposition 3.4,
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 $n\in {\Bbb N}$ ,
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
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
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
Clearly, $\widehat {\mathcal G}^s$ is a subadditive potential on $\Sigma _A$ and
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),
It follows that $P(\Sigma _{A},\sigma ,\widehat {\mathcal {G}}^{s})=0$ .
By [Reference Ban, Cao and Hu2],
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
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
where $G_n, H_n$ are defined in (6.9) and $C=C_{\pmb M}$ . By Proposition 6.7,
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 $x\in \Sigma _A$ ,
where
Hence,
where in the second inequality, we used [Reference Walters39, Theorem 9.7(iv)]. Combining this with (6.10) yields that, for $n\geq N_\epsilon $ ,
so
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
Let $k\in \{0,\ldots , d-1\}$ . Write $u:=\exp (\unicode{x3bb} _{k+1})$ . Then, for m-a.e. $x\in \Sigma _A$ ,
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. $x=(x_n)_{n=1}^\infty \in \Sigma _A$ ,
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,
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. $x=(x_n)_{n=1}^\infty \in \Sigma _A$ ,
It follows that, for m-a.e. $x=(x_n)_{n=1}^\infty \in \Sigma _A$ ,
where in the third inequality we used the Shannon–McMillan–Breiman theorem (cf. [Reference Walters39, 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.
Acknowledgement
The research of Feng was partially supported by the General Research Fund CUHK14301218 from the Hong Kong Research Grant Council, and by a Direct Grant for Research in CUHK. The research of Simon was partially supported by the grant OTKA K104745. The authors are grateful to Ching-Yin Chan for reading an early version of this paper and catching some typos.
A Appendix. Main notation and conventions
For the reader’s convenience, we summarize in Table A1 the main notation and typographical conventions used in this paper.