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.


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 of C 1+α 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 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 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 R d . A finite family {f i } i=1 of contracting self-maps on Z is called a C 1 iterated function system, if there exists an open set U ⊃ Z such that each f i extends to a contracting C 1 -diffeomorphism f i : U → f i (U ) ⊂ U . Let K be the attractor of the IFS, that is, K is the unique non-empty compact subset of R d such that (1.2) (cf. [16]). Let ( , σ ) be the one-sided full shift over the alphabet {1, . . . , } (cf. [7]). Let : → K denote the canonical coding map associated with the IFS {f i } i=1 . That is, with z ∈ U . The definition of is independent of the choice of z. For any compact subset X of with σ X ⊂ X, we call (X, σ ) a one-sided subshift or simply a subshift over {1, . . . , } and let dim S X denote the singularity dimension of X with respect to the IFS {f i } i=1 (cf. Definition 2.5).
For a set E ⊂ R d , let dim B E 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 (X) is bounded above by the singularity dimension of X.  where B(x, r) denotes the closed ball centered at x of radius r. Equivalently, dim P η = inf{dim P F : F is a Borel set with η(F ) = 1}, where dim P F 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. 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 is an affine IFS. Next We refer the reader to [35, §20] for more details. In what follows we always assume that is a repeller of ψ. Let dim S * denote the singular dimension of with respect to ψ (see Definition 6.1). For an ergodic ψ-invariant measure μ on , let dim L * μ be the Lyapunov dimension of μ with respect to ψ (see Definition 6.2). Analogously to Theorems 1.1-1.2, we have the following results. THEOREM 1.3. Let be the repeller of ψ. Then Dimension estimates for C 1 iterated function systems 2677 THEOREM 1.4. Let μ be an ergodic ψ-invariant measure supported on . 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 [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 . To estimate dim B K, 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 = i 1 ...i n f i 1 ...i n (K) and then estimate Under the strong assumption of distortion property, Falconer was able to show that [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 ...i n (B)), and then apply the subadditive thermodynamic formalism to estimate the growth rate of ). 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 ...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, . . . , 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 ...i n (B 0 )) ≤ N 1 . . . 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 ...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 dim B K, say s 1 . Replacing the IFS {f i } i=1 by its nth iteration {f i 1 ...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 , 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 i|n (K)) for m-a.e. (almost every) i ∈ , where u k = exp(λ k ), k = 1, . . . , d, with λ k being the kth Lyapunov exponent of the matrix cocycle A(i, n) := D πσ n i f i|n with respect to m. More precisely, we have the following inequality (see Lemma 5.1): for m-a.e. i ∈ , with the convention that λ 1 + · · · + λ 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 → X a continuous mapping. We call (X, T ) a topological dynamical system. For x, y ∈ X and n ∈ N, we define (2.1) A set E ⊂ X is called (n, ε)-separated if for every distinct x, y ∈ E we have d n (x, y) > ε. Let C(X) denote the set of real-valued continuous functions on X. Let G = {g n } ∞ n=1 be a subadditive potential on X, that is, g n ∈ C(X) for all n ≥ 1 such that g m+n (x) ≤ g n (x) + g m (T n x) for all x ∈ X and n, m ∈ N. (2.2) Following [10], below we define the topological pressure of G. If the potential G is additive, that is, g n = S n g := n−1 k=0 g • T k for some g ∈ C(X), then P (X, T , G) recovers the classical topological pressure P (X, T , g) of g (see, for example, [39]).
Let M(X) denote the set of Borel probability measures on X, and M(X, T ) the set of T-invariant Borel probability measures on X. For μ ∈ M(X, T ), let h μ (T ) denote the measure-theoretic entropy of μ with respect to T (cf. [39]). Moreover, for μ ∈ M(X, T ), by subadditivity we have (2.6) 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 G. There exists at least one ergodic equilibrium measure when the entropy map μ → h μ (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 → X i be continuous. Suppose π : X 1 → X 2 is a continuous surjection such that the following diagram commutes: If, furthermore, there is an integer q > 0 so that π −1 (y) has at most q elements for each y ∈ X 2 , then for each μ ∈ 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]).

Subshifts.
In this subsection we introduce some basic notation and definitions about subshifts.
Let ( , σ ) be the one-sided full shift over the alphabet A = {1, . . . , }. That is, = A N endowed with the product topology, and σ : → is the left shift defined by ( The topology of is compatible with the following metric on : . . x n . Let X be a non-empty compact subset of satisfying σ X ⊂ X. We call (X, σ ) a one-sided subshift or simply a subshift over 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 ∈ N, It is clear that X ⊂ X (n) for every n ∈ N. Let G = {g n } ∞ n=1 be a subadditive potential on a subshift (X, σ ). It is known that in such a case, the topological pressure of G can alternatively be defined by where [i] := {x ∈ : x|n = i} for i ∈ 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.
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 σ -invariance of μ follows directly from its definition, and we only need to prove that h μ ( Without loss of generality we prove this in the case when k = 1. For n ∈ N, let P n denote the partition of A N consisting of the nth cylinders of A N , that is, P n = {[I ] : I ∈ A n }, and set σ −1 P n = {σ −1 ([I ]) : I ∈ A n }. Then it is direct to see that any element in P n intersects at most #A elements in σ −1 P n , and vice versa. Hence, [20,Lemma 4.6]. It follows that This proves the claim.
By the affinity of the measure-theoretic entropy h (·) (σ m ) (see [39,Theorem 8 where the second equality follows from the above claim.

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.
be the one-sided full shift over the alphabet {1, . . . , } and let : → K denote the corresponding coding map defined as in ( For T ∈ R d×d , let α 1 (T ) ≥ · · · ≥ α d (T ) denote the singular values of T. Following [13], for s ≥ 0 we define the singular value function φ s : where k = [s] is the integral part of s.
Definition 2.6. Let m be an ergodic σ -invariant Borel probability measure on . For any i ∈ {1, . . . , d}, the ith Lyapunov exponent of m is The Lyapunov dimension of m with respect to {f i } i=1 , written as dim L m, is the unique non-negative value s for which It follows from the definition of the singular value function φ 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 λ, we have dim L m = h m (σ )/−λ.

Remark 2.7
(i) The concept of singularity dimension was first introduced by Falconer [13,15]; see also [29]. It is also called affinity dimension when the 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.

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, B, m), and let {g n } n∈N be a sequence of L 1 functions satisfying the following subadditivity relation: Suppose that there exists C > 0 such that |g n (x)| ≤ Cn for all x ∈ X and n ∈ N. 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 ∈ N, Since the sequence {g n } is subadditive and satisfies (2.12), by [28, Lemma 2.2], for any 0 < k < n, As a consequence, Notice that, for each f ∈ L 1 and n ∈ 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 lim sup n→∞ E g n n C n ≤ E g k k C 1 almost everywhere for each k, so by the dominated convergence theorem, Combining it with (2.13) yields the desired result lim n→∞ E((g n /n)|C n ) = g almost everywhere.

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.
where X (n) is defined as in (2.7), and P (X (n) , σ n , g n ) denotes the classical topological pressure of g n over the full shift space (X (n) , σ n ).
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.
Proof of Proposition 3.1. We first prove that, for each n ∈ N, To see this, fix n ∈ N and let μ be an equilibrium measure for the potential G. Then ≤ h μ (σ ) + 1 n g n dμ (by (2.5)) where in the last inequality, we use the fact that μ ∈ M(X (n) , σ n ) and the classical variational principle for the topological pressure of additive potentials. This proves (3.2).
In what follows we prove that P (X, σ , G) ≥ lim sup n→∞ 1 n P (X (n) , σ n , g n ). Hence, μ is supported on If x is in this set, then, for each N ≥ 1, there exist integers i(N) and If the values k(N) are unbounded as N → ∞, then (3.6) yields x ∈ X, while if they are bounded then some value of k recurs infinitely often as k(N), which implies that σ k x ∈ X by (3.7). Thus μ is supported on Since σ X ⊂ X, the set (σ −1 X)\X is wandering under σ −1 (i.e., its preimages under powers of σ are disjoint), so it must have zero μ-measure. Consequently, μ ∈ M(X, σ ). 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, σ ) be a one-sided subshift over a finite alphabet A and g, h ∈ C(X). Assume, in addition, that h(x) < 0 for all x ∈ X. Let Set, for 0 < r < r 0 , where X * is the collection of finite words allowed in X and S n h(x) := n−1 k=0 h(σ k x). Then

8)
where t is the unique real number such that P (X, σ , g + th) = 0, and |I | stands for the length of I.
To prove the above result, we need the following lemma.
Dimension estimates for C 1 iterated function systems 2687 LEMMA 3.5. Let (X, σ ) be a one-sided subshift over a finite alphabet A and f ∈ C(X). Then Proof. The result is well known. For the reader's convenience, we include a proof. Define, for n ∈ N, Since f is uniformly continuous, var n f → 0 as n → ∞. It follows that This concludes the result of the lemma since

Proof of Proposition 3.4. Set
for sufficiently small r.

The proof of Theorem 1.1
Recall that, for T ∈ R d×d , α 1 (T ) ≥ · · · ≥ α d (T ) are the singular values of T, and φ s (T ) (s ≥ 0) is defined as in (2.9). We begin with an elementary but important lemma.
Then, for any non-degenerate C 1 map f : U → R d , there exists r 0 > 0 such that, for any y ∈ E, z ∈ B(y, r 0 ) and 0 < r < r 0 , the set f (B(z, r)) can be covered by 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. Since f is C 1 , non-degenerate on U and E is compact, it follows that γ > 0. Take = (2 √ d − 1)/2. Then there exists a small r 0 > 0 such that, for u, v, w ∈ V 2r 0 (E) := {x : and Now let y ∈ E and z ∈ B(y, r 0 ). For any 0 < r < r 0 and x ∈ B(z, r), taking u = x and v = z in (4.2) gives so by (4.3) and (4.1), That is, f (B(z, r)) is contained in an ellipsoid which has principle axes of lengths 4 √ dα i (D y f )r, i = 1, . . . , d. Hence, f (B(z, r)) is contained in a rectangular parallelepiped of side lengths 2 √ dα i (D y f )r, i = 1, . . . , d. Now we can divide such a parallelepiped into at most (D y f )r. Therefore, this parallelepiped (and f (B(z, r)) as well) can be covered by In the remainder of this section let {f i } i=1 be a C 1 IFS on R d with attractor K. Let ( , σ ) be the one-sided full shift over the alphabet {1, . . . , } and : → K the canonical coding map associated with the IFS (cf. (1.3)). As a consequence of Lemma 4.1, we obtain the following proposition.
. Applying Lemma 4.1 to the mappings f i , we see that there exists r 0 > 0 such that, for any y ∈ K, z ∈ B(y, r 0 ), 0 < r < r 0 and i ∈ {1, . . . , }, the set f i (B(z, r)) can be covered by This implies that α 1 (D y f i ) ≤ γ for any y ∈ K and i ∈ {1, . . . , }. 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 ≤ n 0 and i ∈ , that is, the set f i|n (K) can be covered by C 1 n−1 p=0 G(σ p i) balls of radius n−1 p=0 H (σ p i). Below we show by induction that this holds for all n ∈ N and i ∈ .
Suppose, for some m ≥ n 0 , that the conclusion of the proposition holds for any positive integer n ≤ m and i ∈ . Then, for i ∈ , f | (σ i)|m (K) can be covered by C 1 m−1 p=0 G(σ p+1 i) balls of radius m−1 p=0 H (σ p+1 i). Let B 1 , . . . B N denote these balls. We may assume that B j ∩ f | (σ i)|m (K) = ∅ for each j. Since B( σ i, r 0 ). Therefore, f i 1 (B j ) can be covered by θ( σ i, i 1 ) = G(i) balls of radius , it follows that f i|(m+1) (K) can be covered by balls of radius m p=0 H (σ p i). Thus the proposition also holds for n = m + 1 and all i ∈ , as desired.
Next, we provide an upper bound on the upper box-counting dimension of the attractor Let k ∈ {0, 1, . . . , d − 1}. Let G, H : → R be defined as in (4.4). Let t be the unique real number so that Proof. Write g = log G and h = log H for short. Define Then 0 < r min ≤ r max < 1. For 0 < r < r min , define 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 σ -invariant sets under the coding map. PROPOSITION 4.4. Let X be a compact subset of satisfying σ X ⊂ X, and k ∈ {0, . . . , d − 1}. Then, for each n ∈ N, where X (n) is defined as in (2.7), t n is the unique number for which P (X (n) , σ n , (log G n ) + t n (log H n )) = 0, Proof. The result is obtained by applying Proposition 4.3 to the IFS {f I : n stands for the collection of words of length n allowed in X.
We are now ready to prove Theorem 1.1. Hence, where in the second inequality, we used [39, Theorem 9.7(iv)]. Combining this with (4.7) yields that, for n ≥ N , Letting n → ∞ and then → 0, we obtain s ≥ dim B (X), as desired.

The proof of Theorem 1.2
Let : → R d be the coding map associated with a (1.3)). For E ⊂ R d and δ > 0, let N δ (E) denote the smallest integer N for which E can be covered by N closed balls of radius δ. For T ∈ R d×d , let α 1 (T ) ≥ · · · ≥ α d (T ) denote the singular values of T, and let φ 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. G p (σ pi x) ≤ λ 1 + · · · + λ k − kλ k+1 + for large enough n. (5.10) Fix such an x and let p ≥ 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 + ) np in R d can be covered by C 2 (1 + /u) dnp balls of radius u np . It follows that, for large enough n, Hence, by (5. where the first equality follows from the fact that, for pn ≤ m < p(n + 1),

N u m ( ([x|m])) ≤ N u m ( ([x|pn])) ≤ 4 d (u pn−m ) d N u np ( ([x|pn]))
using the fact that, for R > r > 0, a ball of radius R in R d can be covered by (4R/r) d balls of radius r. Letting → 0 yields the desired inequality (5.2).
The following result is also needed in the proof of Theorem 1.2. 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 ∈ N, let n denote the set of the points x = (x n ) ∞ n=1 ∈ such that ( ([x 1 . . . x n ])) .
To prove that (5.11) holds almost everywhere, by the Borel-Cantelli lemma it suffices to show that ∞ n=1 m( n ) < ∞.
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 = λ k+1 < 0). Letting → 0 yields the desired result. 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 , . . . , R } be a Markov partition of with respect to ψ. It is known that this dynamical system induces a subshift space of finite type ( A , σ ) over the alphabet {1, . . . , }, where A = (a ij ) is the transfer matrix of the Markov partition, namely, a ij = 1 if intR i ∩ ψ −1 (intR j ) = ∅ and a ij = 0 otherwise [37], and

Upper bound for the box-counting dimension of C 1 -repellers and the Lyapunov dimensions of ergodic invariant measures
, . . . , } N : a i n i n+1 = 1 for all n ≥ 1}. This gives the coding map : A → such that 2) and the following diagram commutes: (Keep in mind that throughout this section, denotes the coding map for the repeller and no longer for the coding map for an IFS as used in the previous sections.) The coding map is a Hölder continuous surjection. Moreover, there is a positive integer q such that −1 (z) has at most q elements for each z ∈ (see [37, p. 147]).
For n ≥ 1, define Clearly there exists a large number C 1 so that the conclusion of the proposition holds for any positive integer n ≤ n 0 and i ∈ A , that is, the set R i|n can be covered by C 1 n−1 p=0 G(σ p i) balls of radius n−1 p=0 H (σ p i). Below we show by induction that this holds for all n ∈ N and i ∈ A .
Suppose, for some m ≥ n 0 , that the conclusion of the proposition holds for any positive integer n ≤ m and i ∈ A . Then, for given i = (i n ) ∞ n=1 ∈ A , R (σ i)|m can be covered by C 1 m−1 p=0 G(σ p+1 i) balls of radius m−1 p=0 H (σ p+1 i). Let B 1 , . . . , B N denote these balls. We may assume that B j ∩ R (σ i)|m = ∅ for each j. Since so the center of B j is in B( σ i, r 0 ). Therefore, by Lemma 6.3 and (6.5), f i 1 ,i 2 (B j ) can be covered by Since ψ(R i|(m+1) ) ⊂ R (σ i)|m , it follows that hence R i|(m+1) can be covered by balls of radius m p=0 H (σ p i). Thus the proposition also holds for n = m + 1 and all i ∈ A , as desired. PROPOSITION 6.6. Let k ∈ {0, 1, . . . , d − 1}. Let G, H : A → R be defined as in (6.4). Let t be the unique real number so that P ( A , σ , (log G) + t (log H )) = 0.
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 ≤ r max < 1. For 0 < r < r min , define where * A denotes the set of all finite words allowed in A . Clearly {[I ] : I ∈ A r } is a partition of A . By Proposition 6.4, there exists a constant C 1 > 0 such that, for each 0 < r < r min , every I ∈ A r and x ∈ [I ], R I can be covered by It follows that can be covered by This completes the proof of the proposition.
For n ∈ N, applying Proposition 6.6 to the mapping ψ n instead of ψ, we obtain the following result.
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 * ( ). We may assume that s < d; otherwise we have nothing left to prove.
Set k = [s]. Let G s = {g s n } ∞ n=1 be the subadditive potential on defined by g s n (z) = log φ s ((D z ψ n ) −1 ).
Then P ( , ψ, G s ) = 0 by the definition of dim S * ( ). Let G s := { g s n } ∞ n=1 , where g s n ∈ C( A ) is defined by Clearly, G s is a subadditive potential on A and Since the factor map : A → is onto and finite-to-one, by Lemma 2.3, m → m • −1 is a surjective map from M( A , σ ) to M( , ψ) and, moreover, h m (σ ) = h m• −1 (ψ) for m ∈ M( A , σ ). By the variational principle for the subadditive pressure (see Theorem 2.2), It follows that P ( A , σ , G s ) = 0.
Since P ( A , σ , G s ) = 0, the above equalities imply that, for each > 0, there exists N > 0 such that 0 ≤ P ( A , σ n , g s n ) ≤ n for n > N . Hence, where in the second inequality, we used [39, Theorem 9.7(iv)]. Combining this with (6.10) yields that, for n ≥ N , n ≥ − log C + n(s − s n ) log θ, so s ≥ s n + + n −1 log C log θ ≥ dim B + + n −1 log C log θ .
Letting n → ∞ and then → 0, we obtain s ≥ dim B , 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. n log N u n (R x|n ) ≤ (λ 1 + · · · + λ k ) − kλ k+1 , ( 6 . 1 2 ) where N δ (E) is the smallest integer N for which E can be covered by N closed balls of radius δ. 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 * μ < d; otherwise there is nothing left to prove. Since : A → is surjective and finite-to-one, by Lemma 2.3, there exists a σ -invariant ergodic measure m on A so that m • −1 = μ and h m (σ ) = h μ (ψ).
Let u = exp(λ k+1 ) and ∈ (0, 1). Applying Lemma 6.9 (in which we take ρ = u) yields that, for m-a.e. x = (x n ) ∞ n=1 ∈ A , An additional effort is then required to justify that this upper bound is indeed equal to dim S * . 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.
One-sided full shift over the alphabet {1, . . . , } : → K Coding map associated with Lyapunov dimension of m with respect to {f i } i=1 (cf. Definition 2.6) P (X, T , {g n } ∞ n=1 ) Topological pressure of a subadditive potential {g n } ∞ n=1 on a topological dynamical system (X, T ) (cf. Definition 2.1) h μ (T ) Measure-theoretic entropy of μ with respect to T G * (μ) Lyapunov exponent of a subadditive potential G with respect to μ (cf. (2.5)) α i (T ), i = 1, . . . , d The ith singular value of T ∈ R d×d ( §2) φ s Singular value function (cf. (2.9)) S n g g + g • T + · · · + g • T n−1 for g ∈ C(X) G s = {g s n } ∞ Smallest number of closed balls of radius δ required to cover E dim S * Singularity dimension of with respect to ψ (cf. Definition 6.1) dim L * μ Lyapunov dimension of μ with respect to ψ (cf. Definition 6.2)