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Quantitative recurrence properties of expanding maps

Under a map T, a point x recurs at rate given by a sequence {r_n} near a point x_0 if d(T^n(x),x_0)<r_n infinitely often. Let us fix x_0, and consider the set of those x's. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T, the local dimension near x_0 and the upper limit of 1/n log 1/r_n. We apply our results in several concrete examples including subshifts of finite type, Gauss transformation and inner functions.


Introduction
The pre-images under a mixing transformation T distribute themselves somehow regularly along the base space. In this paper we aim to quantify this regularity by studying both the measure and dimension of some recurrence sets. More precisely, we study the behaviour of pre-images under expanding transformations, i.e. transformations which locally increase distances.
Throughout this paper (X, d) will be a locally complete separable metric space endowed with a finite measure λ over the σ-algebra A of Borel sets. We further assume throughout that the support of λ is equal to X and that λ is a non-atomic measure. We recall that a measurable transformation T : X −→ X preserves the measure λ if λ(T −1 (A)) = λ(A) for every A ∈ A. The classical recurrence theorem of Poincaré (see, for example, [23], p.61) says that Here C1 > 0 and 0 < α ≤ 1 are absolute constants. This property allows us to compare the ratio λ(A)/λ(A ) with λ(T (A))/λ(T (A ))for A, A contained in the same block of P0.
Finally, the reason for the name expanding is the property that if x, y belong to the same block of the partition Pn = ∨ n j=0 T −j (P0), then d(T n (x), T n (y)) ≥ C2β n d(x, y) with absolute constants C2 > 0 and β > 1.
In this paper we are interested in studying for expanding systems the size of the set of points of X where lim infn→∞ d(T n (x), x0)/rn = 0, where {rn} is a given sequence of positive numbers and x0 is a previously chosen point in X. To do this we study the size of the set W(x0, {rn}) = {x ∈ X : d(T n (x), x0) < rn for infinitely many n} .
Our first objective is to study the relationship between the measure of this set and how fast goes to zero the sequence of radii. We use the following definitions of local dimension.
Here and hereafter P (n, x) denotes the block of the partition Pn = ∨ n j=0 T −j (P0) which contains the point x ∈ X.
We have the following result. Theorem 1.1. Let (X, d, λ, T ) be an expanding system with finite entropy Hµ(P0) with respect to the partition P0. Let {rn} be a non increasing sequence of positive numbers. Then for λalmost all point x0 ∈ X we have that if ∞ X n=1 r δ n = ∞ for some δ > δ λ (x0), then W(x0, {rn}) has full λ-measure , and we can conclude that, for λ-almost all point x0 ∈ X, lim inf n→∞ d(T n (x), x0) rn = 0 , for λ-almost all x ∈ X .
Further results about the points x0 which satisfy the conclusions in Theorem 1.1 are included in Section 5. There is also a quantitative version when the system has the Bernoulli property (see Theorem 5.2).
If the sequence {rn} tends to zero in such a way that P n λ(B(x0, rn)) < ∞ then it is easy to check that λ(W(x0, {rn})) = 0 and therefore lim inf n→∞ d(T n (x), x0) rn ≥ 1 , for λ-almost all x ∈ X .
Even in the case that the set W(x0, {rn}) has zero λ-measure we have proved that this set is large since we have obtained a positive lower bound for its dimension.
In this paper we use two different notions of dimension: the λ-grid dimension (Dim Π,λ ), considering coverings with blocks of the partitions Pn, and the λ-Hausdorff dimension (Dim λ ), when we consider coverings with balls of small diameter (see Section 2 for the definitions). We remark that Dim λ is equal to 1/N times the usual Hausdorff dimension when λ is the Lebesgue measure in X = R N . To obtain lower bounds for the dimension, we have constructed a Cantor-like set contained in W(x0, {rn}). The elements of the different families of the Cantor set are certain blocks of some partitions Pn. In our construction these blocks have controlled µ-measure and they are in a certain sense well distributed. The main difficulty while estimating the dimension of this Cantor set is that does not have a fixed pattern and the ratio between the measure of a 'parent' and his 'son' can be very big depending on the sequence of radii. Our approach is contained in Theorem 2.1.
The main tools in the construction of the Cantor set are: (1) good estimates for the measure µ of some blocks of Pn, obtained as a consequence of Shannon-McMillan-Breimann Theorem (see Theorem D); (2) good estimates of the ratio between λ(P (n + 1, x)) and λ(P (n, x)) due to the distortion property of J. An extra difficulty is that the measures λ and µ are only comparable in each block of the partition P0.
In order to obtain lower bounds for Dim λ we relate it with Dim Π,λ and to do so we have required an extra condition of regularity over the 'grid' Π = {Pn} (see Section 2). The required condition is trivially fulfilled in the one dimensional case. When the measure λ of a ball is comparable to a power of its diameter we have also obtained an estimate of Dim λ without assuming the regularity condition on the grid. 1. If the grid Π is λ-regular then 1 Dim λ (W(U, x0, {rn})) − 1 ≤ δ λ (x0) hµ .
2. If λ is a doubling measure verifying that λ(B(x, r)) ≤ C r s for all ball B(x, r), then As a consequence, we obtain the same estimates for the Hausdorff dimensions of the set Observe that, for instance, if X ⊂ R, we obtain that, for any α > 0, Dim λ {x ∈ U : lim inf n→∞ e nα d(T n (x), x0) = 0} ≥ hµ hµ + δ λ (x0) α .
Theorem 1.2 is sharp since for some expanding systems we get equality, see Theorem 7.4. As in Theorem 1.1 we have chosen to state the above result for almost all x0 ∈ X and we refer to Section 6.1 for more precise results concerning to the set of points x0 where this kind of results holds.

Coding.
It is a well known fact that an expanding map induces a coding on the points of X (see Section 4.1). Via this coding the above results are in certain sense a consequence of analogous results involving symbolic dynamic. More precisely, each point x of the set Given an increasing sequence {t k } of positive integers and a point x0 ∈ X0 we study the size of set f W(x0, {tn}) = {x ∈ X0 : T k (x) ∈ P (t k , x0) for infinitely many k}.
If x ∈ X0 and T k (x) ∈ P (t k , x0) then P (j, T k (x)) = P (j, x0) for j = 0, 1, . . . , t k and it follows that f W(x0, {tn}) can be also described as Moreover, if the partition P0 is finite or if the system has the Bernoulli property, (i.e. if T (P ) = X (mod 0) for all P ∈ P0), then we have the following quantitative version: lim n→∞ #{i ≤ n : T i (x) ∈ P (ti, x0)} P n j=1 µ(P (tj, x0)) = 1 , for λ-almost every x ∈ X , where µ is the ACIPM associated to the system.
Property (3) is related to the decay of the correlation coefficients of the indicator functions of {P (n, x0)}, see [40] and [35]. For expanding systems with the Bernoulli property L.S. Young [44] has proved that this decay is exponential.
As with W(x0, {rn}), it is easy to see using the Borel-Cantelli lemma that if P n λ(P (tn, x0)) < ∞, then λ( f W(x0, {tn})) = 0. As in the previous theorems, for the sake of simplicity, we have stated this last result for almost all point x0 in X, but we refer to Section 6.1 for a more precise statement concerning the points x0 which satisfy the conclusions. Also, we should mention that, up a λ-zero measure set, we have that L(x0)/hµ = lim sup n→∞ tn/n (see Theorem 6.3).

Applications.
The generality of the definition of expanding systems allows us to apply our results in a broad kind of situations. In the final section, we have obtained results for Markov transformations, subshifts of finite type (in particular, Bernoulli shifts), the Gauss transformation, some inner functions and expanding endomorphisms. In the case of Bernoulli shifts we also give a precise upper bound of the dimension by using a large deviation inequality. As an example, for the Gauss map φ, which acts on the continued fractions expansions as the left shift, we have the following results: (1) If α > 1 then, for almost all x0 ∈ [0, 1], and more precisely, if x0 is an irrational number with continued fraction expansion [ i0 , i1 , .
for infinitely many n.
The techniques developed in this paper for expanding maps and therefore for one-sided Bernoulli shifts, can be extended to bi-sided Bernoulli shifts. This has allowed us [20] to get results on recurrence for Anosov flows.
The outline of the paper is as follows: In Section 2 we give our two definitions of dimension and prove some general results for computing the dimensions of a kind of Cantor-like sets with the particular feature that the ratio between the size of a 'son' and his 'parent' decays very fast. Section 3 contains some consequences of Shannon-McMillan-Breiman Theorem. In section 4 we give the complete definition of an expanding system and in Section 4.1 we recall how to associate a code to the points of X. In Section 4.2 we prove some general properties of expanding maps. The precise statements and proofs of Theorems 1.1 and 1.3, and some consequences of them, are contained in Section 5. The dimension results are included in Section 6. More general versions of Theorems 1.2 and 1.4 are included in Section 6.1. In Section 6.2 we include an upper bound of the dimension. Finally, Section 7 contains several applications of the above results.
Acknowledgements: We want to thank to J. Gonzalo, R. de la Llave, V. Muñoz and R. Pérez Marco for helpful conversations about this work. We are particularly indebted to A. Nicolau for his encouragement and stimulating discusions.
A few words about notation. There are many estimates in this paper involving absolute constants. These are usually denoted by capital letters like C. Occasionally, we shall indicate a constant C depending on some parameter α by C(α). The symbol #D denotes the number of elements of the set D. By A B we mean that there exist absolute constants C1, C2 > 0 such that C1B ≤ A ≤ C2B.

Grids and dimensions.
Along this section (X, d, A, λ) will be a finite measure space with a compatible metric. Compatible means that A is the the σ-algebra of the Borel sets of d. We recall that we are assuming that the measure λ is non-atomic and its support is X.
where the infimum is taken over all the coverings {Bi} of F with balls such that diam (Bi) ≤ ε for all i.
It is not difficult to check that H α is a regular Borel measure, see e.g. [32]. Observe that if X ⊂ R N and λ is Lebesgue measure, then H α is comparable with the usual N α-dimensional Hausdorff measure.
Definition 2.2. The λ-Hausdorff dimension of F is defined as If X ⊂ R N and λ is Lebesgue measure, then the λ-Hausdorff dimension coincides with 1/N times the usual Hausdorff dimension.
In order to compute the λ-grid Hausdorff dimension we will use the following result which parallels Frostman lemma. Lemma 2.1. Let Π = {Pn} be a grid of X. For each n ∈ N, let Qn be a subcollection of Pn and let F be a set with If there exist a measure ν such that ν(F ) > 0, a real number 0 < γ ≤ 1 and a positive constant where Q(k, x) denotes the block of Q k which contains x, then, Proof. It follows from the fact that H α Π,λ (F ) ≥ ν(F )/C > 0.
The following result allows to obtain a lower bound for the λ-grid Hausdorff dimension of Cantor-like sets. (iii) For each e Jj ∈ e Jj (j ≥ 1) there exists a unique Jj−1 ∈ Jj−1 such that e Jj ⊂ Jj−1.
Let C be the Cantor-like set defined by Assume that the pattern of C has the following additional properties: (1) There exist two sequences {αj} and {βj} of positive numbers such that αj ≤ λ( e Jj) λ(Jj−1) ≤ βj .
(4) There exists an absolute constant Λ such that for all j large enough Then Remark 2.2. Observe that in the special case when the two families e Jj and Jj coincide and αj = α, βj = β, δj = δ, then the above result is the usual Hungerford's Lemma (Λ = log(β/δ)/ log α), see e.g. [37].
Proof. We construct a probability measure ν supported on C in the following way: We define ν(J0) = 1 and for each set Jj ∈ Jj we write where Jj−1 and e Jj denote the unique sets in Jj−1 and e Jj respectively, such that Jj ⊂ e Jj ⊂ Jj−1. As usual, for any Borel set B, the ν-measure of B is defined by where the infimum is taken over all the coverings U of B ∩ C with sets in S Jj. We will show that there exists a positive constant C such that for all x ∈ C and m large enough, and therefore, from Lemma 2.1, we get the result.
To prove (4) let us suppose first that P (m, x) = Jj for some Jj ∈ Jj. From properties (1)-(3) we have that and follows from property (4) that This condition is stronger than (4) for e Jj and Jj and we will use it to get (4) in general. Now, let us suppose that P (m, x) = Jj for all j and for all Jj ∈ Jj. Since x ∈ C there exist Jj ∈ Jj and Jj+1 ∈ Jj+1 such that Jj+1, then from the definition of ν and (5) for e Jj+1 we get And using property (3) and (5) we obtain that But λ(Jj)) ≥ λ(P (m, x)) and so we get ν(P (m, x)) ≤ C (λ(P (m, x))) Λ .
The next definition states some kind of regularity on the distribution of the blocks of the partitions. This property will allow us to relate the Hausdorff dimension with the grid Hausdorff dimension.
Definition 2.5. Let Π = {Pn} be a grid of X. We will say that Π is λ-regular if there exists a positive constant C such that for all ball B λ(∪{P : for all n such that sup P ∈Pn λ(P ) ≤ λ(B).

Remark 2.4.
It is clear from the definition that any grid of X ⊂ R is λ-regular (we can take C = 3) An example: Let X be the square [0, 1] × [0, 1] in R 2 and let us denote by λ the Lebesgue measure. Consider the grid Π = {Pn} defined as follows: the elements of P0 are the four open rectangles obtained by dividing the square [0, 1] × [0, 1] through the lines x = a and y = b, with 1 2 < b < a < 1; the elements of Pn are getting by dividing each rectangle of Pn−1 in four rectangles using the same proportions. We will see that this is not a regular grid.
Let us consider the ball B k with diameter (1 − b) k and contained in the square [0, It is easy to see that sup P ∈Pn λ(P ) = (ab) n , and therefore (ab) n ≤ λ(B k ) implies Therefore, if Π is regular, then for n = n(k) = 2k log(1/(1−b)) log 1/(ab) + C the quotient has to be bounded. But, it is easy to see that the elements of Pn whose closure intersects to [0, 1] × {0} are rectangles of width a n , and hence, since b < a, Therefore, this grid is not regular. On the other hand it is clear that any grid in X whose elements are all squares is regular.
The following result gives a lower bound for the Hausdorff dimension of Cantor like sets which are constructed using a regular subgrid with some control into the quotient between the size of parents and sons. Then, for any subset F ⊆ T n S Q∈Qn Q, for all α and η such that where C is an absolute positive constant. In particular, Proof. Let us consider a ball B such that B ∩ F = ∅ and let n = n(B) be the smallest integer such that bn ≤ λ(B). Then bn ≤ λ(B) < bn−1.
We denote by Q(B) the collection of elements in Qn whose intersection with B is not empty. Then the collection Q(B) is a covering of B ∩ F , that is and moreover by the Definition 2.5 and the election of n = n(B), We may assume that n is large because diam (B) is small and so the above inequality and (8) imply that X The inequality (7) follows now from (10) and (11). To prove (9) let us observe that we can assume that lim sup n→∞ log(1/an) log(1/bn−1) since in other case (9) is trivial. Let us choose now α and η such that lim sup n→∞ log(bn−1/an) log(1/bn) Then H α Π,λ (F ) > 0 and by (7) we have also that H 1−(1−α)η λ (F ) > 0. Since α y η are arbitrary numbers verifying (12), the ineguality (9) follows.

Some consequences of Shannon-McMillan-Breiman Theorem
Along this section (X, A, µ) will be a finite measure space and T : X −→ X will be a measurable transformation. A partition of X is a family P of measurable sets with positive measure satisfying 1. If P1, P2 ∈ P then µ(P1 ∩ P2) = 0.
It follows from these properties that P must be finite or numerable. The entropy of a partition P is defined as If T : X −→ X preserves the measure µ, then the entropy of T with respect to the partition P is This limit exists since the sequence in the right hand side is decreasing. Hence hµ(T, P) ≤ Hµ(P). Finally, the entropy hµ(T ) of the endomorphism T is the supremum of hµ(T, P) over all the partitions P of X with entropy hµ(T, P) < ∞.
If the partition P is generating, i.e. if ∨ ∞ j=0 T −j (P) generates A, then, by the Kolmogorov-Sinai Theorem ([M, p. 218-220]), we get Theorem C Let (X, A, µ) be a probability space and T : X −→ X be a measure preserving transformation. If P is a generating partition of X and the entropy Hµ(P) is finite, then hµ(T ) = hµ(T, P).
Let P (n, x) denotes the element of the partition ∨ n j=0 T −j (P) which contains the point x ∈ X. It follows from the definition of partition, that for almost every x ∈ X, P (n, x) is defined for all n. Entropy is a measure of how fast µ(P (n, x)) goes to zero. The following fundamental result, which is due to Shannon, McMillan and Breiman, formalizes this assertion: Theorem D([M, p. 209]) Let (X, A, µ) be a probability space and let T : X −→ X be a measure preserving ergodic transformation. Let P be a partition with finite entropy Hµ(P). Then, lim n→∞ 1 n log 1 µ(P (n, x)) = hµ(T, P) , for µ-almost every x ∈ X.
We will need later the following consequence of Theorem D.
Lemma 3.1. Let (X, A, µ) be a probability space and let T : X −→ X be a measure preserving ergodic transformation and P be a partition with finite entropy Hµ(P). Then, given ε > 0 there exists a decreasing sequence of sets {E ε N } N ∈N such that and for all x ∈ X \ E ε N e −j(hµ+ε) < µ(P (j, x)) < e −j(hµ−ε) , for all j ≥ N .
with hµ = hµ(T, P) the entropy of T with respect to µ and the partition P.
Proof. Given ε > 0 we define for all j ∈ N the sets By Theorem D we know that for almost every x ∈ X lim n→∞ 1 n log 1 µ(P (n, x)) = hµ .
Therefore there is a set S with µ(S) = 0 such that for all x ∈ X \ S there exists n(x) ∈ N such that˛1 j log 1 µ(P (j, x)) − hµ˛< ε for all j ≥ n(x) . Hence, We define By definition E ε N +1 ⊂ E ε N for all N ∈ N and by (15) Moreover, if x ∈ X \ E ε N , then x ∈ F ε j for all j ≥ N , and therefore e −j(hµ+ε) < µ(P (j, x)) < e −j(hµ−ε) for all j ≥ N .
Proposition 3.1. Let (X, A, µ) be a probability space, let T : X −→ X be a measure preserving mixing transformation, and P be a partition with finite entropy Hµ(P). Let us denote Let P1, P2 be two fixed elements of P.
Proof. We have that

Expanding maps.
We will say that (X, d, A, λ, T ) is an expanding system if (X, A, λ) is a finite measure space, λ is a non-atomic measure and the support of λ is equal to X, (X, d) is a locally complete separable metric space, A is its Borel σ-algebra and T : X −→ X is an expanding map, i.e. a measurable transformation satisfying the following properties: (A) There exists a collection of open sets P0 = {Pi} of X such that sup   and moreover there exist absolute constants 0 < α ≤ 1 and C1 > 0, such that for all Then, there exist absolute constants β > 1 and C2 > 0 such that for all x, y in the same element of Pn we have that d(T n (x), T n (y)) ≥ C2β n d(x, y) . 3. As a consequence of property (C) we have, since sup P ∈P 0 diam (P ) < ∞, that Therefore (see for example [M, p.13]) we deduce that the partition P0 is generating. This means that A = W ∞ n=0 Pn (mod 0). We will define also, for each n ∈ N, the function Then it follows easily that and, in particular, for each measurable set A contained in some element of Pn−1.
Notice that in the definition of an expanding map the measure λ it is not required to have special dynamical properties. However it is a remarkable fact that it is possible to find an invariant measure which is essentially comparable to λ and has very interesting dynamical properties. More concretely it is known the following result Theorem E ([M, p.172]). Let (X, d, A, λ, T ) be an expanding system. Then, there exist a unique probability measure µ on A which is absolutely continuous with respect to λ and such that (i) T preserves the measure µ.
(iii) For each Pi ∈ P0 there exist a positive constant Ki such that (iv) T is exact with respect to µ.
In what follows we will refer to µ as the ACIPM measure associated to the expanding system.  T˛Pi must be injective for all Pi ∈ P0. If we strengthen this condition by requiring also that inf P ∈P 0 λ(T (P )) > 0 and sup or, in particular, if T : P −→ X is bijective (mod 0) for all P ∈ P0 and X is bounded, then, a slight modification of the proof of Theorem E in [30] (using Remark 4.6 instead of [M, Lemma 1.5]), allows to obtain the property (iii) of µ with an absolute constant K. Therefore with this additional assumption one have that Of course, this condition also holds if the partition P0 is finite.
For expanding maps there exists an alternative way of computing the entropy of T : Theorem F([M, p. 227]) Let (X, d, A, λ, T ) be an expanding system and let µ be the ACIPM measure associated to the system. If the entropy Hµ(P0) of the partition P0 is finite, then log J is integrable and

A code for expanding maps
We will denote by P (n, x) the element of the collection Pn which contains the point x ∈ X. Observe that for each n, P (n, x) is well defined for x belonging to Υn := ∪{P : P ∈ Pn} and Υn has full λ-measure for property (A.3) for Pn, see Remark 4.1.1. Therefore if x belongs to the set then P (n, x) is well defined for all n. Moreover, if x ∈ Υn then from the definition of Pn we have that for all ∈ N, and so P (n, T (x)) is well defined for all n, ∈ N. This set has full λ-measure since X \ X0 ⊆ ∪ n≥0 X \ Υn and this set has zero λ-measure by (A.3) for Pn. Hence, for almost every x ∈ X, P (n, T (x)) is defined for all n, ∈ N. An easy consequence of the definition of P (n, x) that we will use in the sequel is that If x ∈ X0 then, since P (n + 1, x) ⊂ P (n, x) and diam (P (n, x)) → 0 when n → ∞, we have that \ n P (n, x) = {x} and so the sequence {P (n, x)}n determines to the point x. Moreover, from (19) we have that T n (P (n, x)) = P (0, T n (x)), and it is not difficult to see that and the sequence {P (0, T n (x))}n also determines the point x.
We will also define the set X + 0 as the union of X0 with the set of points x ∈ X verifying that there exists a sequence {Pn}, with Pn ∈ Pn and Pn+1 ⊂ Pn, such that We remark that for points x ∈ X + 0 \ X0 the sequence {Pn} is not uniquelly determinated by x. From now on, for each x ∈ X + 0 \ X0 we make an election of {Pn} and we denote Pn by P (n, x). Also by P (0, T n (x)) we mean T n (P (n, x)). We are extending in this way the definition of P (n, x) and P (0, T n (x)) given for points in X0 in such a way that for points in X + 0 \ X0 we also have that T n (P (n, x)) = P (0, T n (x)).

Some properties of expanding maps
Let (X, d, A, λ, T ) be an expanding system. Following [30] we have Proposition 4.1. There exists an absolute constant C > 0 such that for all x0 ∈ X + 0 and for all natural number n we have that if x, y ∈ P (n, x0) then Moreover, if sup P ∈P 0 diam (T (P )) < ∞, then (20) holds for s = n + 1.
Proof. We will prove the lemma for the case s = n+1. If x, y ∈ P (n.x0) we have, from properties (B) and (C), that where D = sup P ∈P 0 diam (P ) and D = sup P ∈P 0 diam (T (P )).
An easy consequence of the above bound, that we will often use, is the following one: If P is an element of Pn, i.e. if P = P (n, x) for some x ∈ X, and P is a measurable subset of P then with C an absolute constant. Moreover, if sup P ∈P 0 diam (T (P )) < ∞, then the above inequality is true for j = n + 1.
Proof. Using (17) we get and the result follows from Proposition 4.1.
Lemma 4.1. If A ∈ A and Q ∈ Pm for some m, then Proof. We may assume that T − (A) ∩ Q = ∅ and A ⊆ P ∈ Pm. The general result follows from the fact that Pm is a partition of X. Then we have that T − (A) ∩ Q is a union of some elements B1, B2, . . . such that Bi ⊂ Pi ∈ P +m and T ˛B j : Bj −→ A is bijective for all j. Let us denote by Sj its inverse map, Sj : A −→ Bj. Then But using (17) we deduce that Therefore, since T (Sj(A)) = A, we have If we denote, for each j, y = Sj(x) we have that y ∈ T − (x) ∩ Bj and that y is unique. This observation completes the proof.
with C > 0 a constant depending on P0.
Proof. Using (19), (17) and Proposition 4.1 we deduce that If ≥ m we have that P ( , y) ⊂ Q for all y ∈ T − (x) ∩ Q and so On the other hand, if < m the map T is injective in Q and therefore there is at most one point y ∈ T − (x) ∩ Q. Since P ( , y) ⊃ P (m, y) = Q we also have that P ( , z) = P ( , y) for any z ∈ Q. Therefore, in this case, the result follows from (21).
with C a constant depending on C0 and D0.
Proof. Notice that from Proposition 4.1 we get that Therefore, 1 J (y) The rest of the proof is similar to the proof of Lemma 4.2.
Proposition 4.3. Let µ be the ACIPM measure associated to the expanding system. Let A ∈ A and Q ∈ Pm with A, Q ⊂ P0 ∈ P0. Then, we have that where z is any point of Q and C > 0 is a constant depending on P0.
Proof. By Theorem E we know that and the result is a consequence of lemmas 4.1 and 4.2.
Remark 4.7. If infP ∈P 0 λ(T (P )) > 0 and sup P ∈P 0 diam (T (P )) < ∞, then λ and µ are comparable in the whole X and it is not necessary in the statement of the Proposition 4.3 that A, Q ⊂ P0.
Recall now the definitions of lower and upper P-dimensions, see Definition 1.1. Since the sequence {P (n, x0)} n∈N is defined for all x0 ∈ X + 0 , we have that δ λ (x0) and δ λ (x0) are also defined for x0 ∈ X + 0 , with β > 1 the constant in the property (C) of expanding maps.
Another quantity that we will need is the following: The rate of decay of the measure λ at x ∈ X0 with respect to the partition P0 is defined as .
Notice that we can extend the definition of τ λ (x0) to all x0 ∈ X + 0 . Lemma 4.4. If the entropy Hµ(P0) of the partition P0 with respect to the ACIPM measure associated to the expanding system is finite, then the set of points x0 ∈ X + 0 verifying In particular, if the partition P0 is finite, then (23) holds for all x0 ∈ X + 0 .
Proof. By Definition 4.2 we have that for n large enough Hence, for n large enough, But from the property (C) of expanding maps, if x0 ∈ X0 we have that, , we obtain the same conclusion since P (n−1, x0) = P (n − 1, x) for x ∈ P (n − 1, x0) ∩ X0. Therefore, in any case, we get that with C > 0. Finally from the definition of δ λ (x0) we have that for n large The result follows from (24), (25), and (26).
Using lemma 3.1 we can define an important subset of X0 which also has full λ-measure. We will refer to this set in the rest of the paper. The following lemma summarizes its properties. Lemma 4.6. Let (X, A, λ, T ) be an expanding system such that the entropy Hµ(P0) of the partition P0, with respect to the unique T -invariant probability measure which is absolutely continuous with respect to λ, is finite. Let X1 denote the subset of X0 with {E 1/m N } the sets given by Lemma 3.1 for ε = 1/m. Then λ(X1) = λ(X) and moreover, if x0 ∈ X1 then: (ii) For all positive integer m there exists N ∈ N such that for all n ≥ N 1 M e −n(hµ+1/m) < λ(P (n, x0)) < M e −n(hµ−1/m) (27) with M > 0 depending on P (0, x0).
The property (i) is satisfied for all points in X0 and therefore also in X1. If the point x0 ∈ X1, then, for all positive integer m, x0 does not belong to ∩N E 1/m N , and so from Lemma 3.1 we have that for all n large enough e −n(hµ+1/m) < µ(P (n, x0)) < e −n(hµ−1/m) .
From part (iii) of Theorem E, we conclude that (27) holds.
Moreover, from (27) we also get that Therefore by taking m → ∞, we get the property (iii).

Measure results
We want to study the size of the set where {rn} is a given sequence of positive numbers and x0 is an arbitrary point in X. Observe that if the sequence {rn} is constant this set is T -invariant, but, in general, this is not the case. We are also interested in the size of another set that we will see that is closely related with W(x0, {rn}). This set is x0) for infinitely many k} with {t k } an increasing sequence of positive integers and x0 ∈ X + 0 .
Notice also that if x ∈ f W(x0, {tn}) then P (m, x) is well defined for infinitely many m. and so it is well defined for all m. Therefore f W(x0, {tn}) ⊂ X0.

Let us denote
With these notations, we have that An .
The following result on the size of these sets is a consequence of the direct part of Borel-Cantelli lemma and Theorem E.
i) Let x0 ∈ ∪P ∈P 0 P and let {rn} be a sequence of positive numbers.
ii) Let x0 ∈ X + 0 and let {tn} be a non decreasing sequence of positive integers.
Proof. i) First of all, let us observe that limn→∞ rn = 0. Let µ be the ACIPM associated to the system. We have that µ(B(x0, r k )) = µ(A k ) for all k ∈ N and therefore since for rn small enough, B(x0, rn) ⊂ P (0, x0) and λ and µ are comparable in that set by Theorem E. From Borel-Cantelli lemma it follows that µ(W(x0, {rn})) = 0 and using the Remark 4.2, we conclude that λ(W(x0, {rn})) = 0. The same argument works for part ii).
Corollary 5.1. Let (X, d, A, λ, T ) be an expanding system. Let x0 ∈ ∪P ∈P 0 P and let {rn} be a sequence of positive numbers. If Corollary 5.2. Let (X, d, A, λ, T ) be an expanding system. Let x0 ∈ ∪P ∈P 0 P such that and let {rn} be a sequence of positive numbers such that If there exists a constant ∆(x0) such that λ(B(x0, r)) ≤ Cr ∆(x 0 ) for all r small enough, then the conclusion holds when Proof. By definition of ∆ λ (x0), we have that for any r small enough Now, for any m ∈ N, since limn→∞ rn = 0, we have that for n big enough, (depending on m), Therefore, for all m ∈ N, From Corollary 5.1 we get that, for all m ∈ N, The result follows now from the fact that Remark 5.1. If the measures λ and µ are comparable in X, then part i) of Proposition 5.1 and its corollaries hold for all x0 ∈ X. In particular, this happens if inf P ∈P 0 λ(T (P )) > 0 and sup see Remark 4.3. Also, part i) of Proposition 5.1 and its corollaries hold for those x0 such that the set of elements P ∈ P0 such that x0 belongs to ∂P is finite. For example, if X ⊆ R or if the partition P0 is finite, then all x0 ∈ X satisfy the above condition.
Theorem 5.1. Let (X, d, A, λ, T ) be an expanding system. Let x0 be a point of X + 0 such that δ λ (x0) > 0 and let {tn} be a non decreasing sequence of positive integers numbers.
Moreover, if the partition P0 is finite or if the system has the Bernoulli property, (i.e. if T (P ) = X (mod 0) for all P ∈ P0), then we have the following quantitative version: In the proof of Theorem 5.1 we will use the following classical result.
Proof of Theorem 5.1. Let µ be the ACIPM associated to the system. For j ≥ k, we have that and by using Proposition 4.3 with = j − k, n = tj and m = t k , and using again that T preserves the measure µ, we conclude that with C > 0 depending on P (0, x0). Let us denote by Zn and Z the counting functions If we compute the expectation value of Z 2 n (with respect to µ), we obtain and using (28) we get Since by Theorem E the measures λ and µ are comparable in P (0, x0) we get from Lemma 4.3 that with C a positive constant. From (29), and (30) we obtain that By applying Paley-Zygmund Lemma we obtain from (31) that Using again that λ and µ are comparable in P (0, x0), we get that and from the hypothesis of the theorem, we obtain that E(Zn) → ∞ as n → ∞. Hence, we have from (32) that and we conclude that f W(x0, {tn}) has positive µ-measure. If we denote, for each n ∈ N f it is easy to see that and since T is exact with respect to µ (see Theorem E) it follows that f W(x0, {tn}) has full µ-measure. Therefore from Remark 4.2 we conclude that f W(x0, {tn}) has full λ-measure. Finally, if the system has the Bernoulli property then the correlation coefficients of the sets {P (n, x0)} n∈N have exponential decay, see [44]. Concretely, she proves that |µ(T − (P (n, x0)) ∩ P (m, x0)) − µ(P (n, x0))µ(P (n, x0))| ≤ C µ(P (n, x0)) e −α (33) for some absolute positive constants C and α and for all m, n, ∈ N. The same argument used in the proof of Theorem 1 in [19], gives the quantitative version.
If the partition P0 is finite, then the dynamical system (X, A, µ, T ) is isomorphic via coding to a (one-sided) subshift of finite type. The stochastic matrix M of this subshift is defined in the following way: pi,j = µ(T −1 (Pj) ∩ Pi)/µ(Pi) where Pi, Pj ∈ P0. Property (A.7) implies that M verifies that M n 0 has all its entries positive, see for example [28], p.158 or Lemma 12.2 in [30]. This implies that the shift σ is mixing, see for example Proposition 12.3 in [30], and moreover (33) follows from the Perron-Frobenius theorem (see, for example [28] or [30]; see also [10]).
We state now the following corollary of this proof. Corollary 5.3. Let (X, d, A, λ, T ) be an expanding system with finite entropy Hµ(P0) with respect to the partition P0 where µ is the ACIPM associated to the system. Let {tn} be a non decreasing sequence of positive integers.
Then for λ-almost all point x0 ∈ X + 0 , more concretely if x0 ∈ X1 (see definition in Lemma 4.6), we have that Proof. If x0 ∈ X1 we have from Lemma 4.6 that for all m ∈ N there exists N ∈ N such that e −n(hµ+1/m) ≤ µ(P (n, x0)) ≤ e −n(hµ−1/m) for all n ≥ N . Therefore we can substitute the inequality (30) by since hµ > 0 (see Remark 4.4) and we can take m large enough so that 0 < 1/m < hµ. Hence we do not need now the hypothesis δ λ (x0) > 0.
Theorem 5.2. Let (X, d, A, λ, T ) be an expanding system. Let x0 be a point of X + 0 such that and let {rn} be a non increasing sequence of positive numbers.
Moreover, if the partition P0 is finite or if the system has the Bernoulli property, (i.e. if T (P ) = X(mod 0) for all P ∈ P0), then we have the following quantitative version: with C a positive constant depending on x0 and on the comparability constants between λ and µ at P (0, x0). for some absolute positive constants C and α and for all n, ∈ N, then using the same arguments that in Theorem 1 in [19] we would have Remark 5.3. We recall that by Lemma 4.4 we know that τ λ (x0) = 0 for λ-almost all x0 ∈ X.
We have also that τ λ (x0) = 0 if infj λ(P (0, T j (x0))) > 0 and sup P ∈P 0 diam (T (P )) < ∞. In particular, if the partition P0 is finite and sup P ∈P 0 diam (T (P )) < ∞, then τ λ (x0) = 0 for all x0 ∈ X + 0 . Corollary 5.4. Under the same hypothesis than Theorem 5.2, if then, Proof. From Theorem 5.2 it follows easily that if the radii rn verify (34) then But notice that for any m ∈ N the radii rn/m also verify (34) and therefore we get that for any The result follows now from the fact that Proof of Theorem 5.2. Let µ be the ACIPM associated to the system. Given x0 ∈ X + 0 and the sequence r k we define t k as the smallest integer so that Hence, f W(x0, {tn}) ⊂ W(x0, {rn}). Moreover, since δ λ (x0) < ∞ and τ λ (x0) < ∞, from Lemma 4.5, we get that But from the definition of t k we have that and since x0 ∈ P (t k − 1, x0) we can conclude that diam(P (t k − 1, x0)) ≥ r k .
Hence, the quantitative version follows from Theorem 5.1.

Lower bounds for the dimension
Our lower estimate of the dimension is based into the construction of a Cantor like set. Our argument requires to compare the measures λ and µ several times because we use some consequences of the Shannon-McMillan-Breiman Theorem for the measure µ (see Section 3) and also some consequences of the definition of expanding maps involving the measure λ. We have already mentioned that the measures λ and µ are comparable into the blocks of the partition P0, but in order to control the comparability constants in our proof, we need the following definition: Definition 6.1. We will say that a point x0 ∈ X0 is approximable if there exist an increasing sequence I(x0) = {pi} of natural numbers such that for all A ∈ A contained in P (0, T p i (x0)) for some i, we have that with K > 1 a constant depending on x0.
Remark 6.1. A mixing version of Poincare's Recurrence Theorem (see [19], Theorem A') shows that for λ-almost all point x0 there exists an increasing sequence {pi} such that P (0, T p i (x0)) = P (0, T p 1 (x0)) for all i. Therefore, the set of approximable points have full λ-measure.
Remark 6.2. From part (iii) of Theorem E we have that if the partition P0 is finite then any point in X + 0 is an approximable point. More generally, from Remark 4.3 we have that if infP ∈P 0 λ(T (P )) > 0 and sup P ∈P 0 diam (T (P )) < ∞, then any point in X + 0 is an approximable point.
The next theorem contains a lower bound for the Hausdorff and the grid Hausdorff dimensions of W(U, x0, {rn}) with respect to the grid Π = {Pn}. As we mentioned in Section 2, in order to get results for the λ-Hausdorff dimension we need an extra property of regularity. More precisely, we ask Π to be λ-regular (see Definiton 2.5). We recall that any grid on R is λ-regular.
where = lim sup n→∞ 1 n log 1 rn and hµ is the entropy of T with respect to µ. Moreover, for all approximable point x0 ∈ X0, the Hausdorff dimensions of the set W(U, x0, {rn}) verify: 1. If the grid Π is λ-regular then 2. If λ is a doubling measure verifying that λ(B(x, r)) ≤ C r s for all ball B(x, r), then Here β is the constant appearing in the property (C) of expanding maps.
First, let us observe that the λ-Hausdorff dimension and µ-Hausdorff dimensions coincide for subsets of ∪P ∈P 0 P and, in particular, for subsets of X0. and Proof. We will prove only the equality of grid-dimensions, since the other proof is similar. By properties (A.2) and (A.3) of expanding maps we have for the α-dimensional λ-grid and µ-grid Hausdorff measures that where P0 = {Pi}. As a consequence of part (iii) of Theorem E we get that with constants depending on i. Therefore Proof of Theorem 6.1. We may assume that δ λ (x0), τ λ (x0) and are all finite, since otherwise the estimations (37) and (38) are trivial. Since µ(U ) > 0, the set U contains a point x ∈ X0.
As U is open, we have that there exists r > 0 such that B(x, r) ⊂ U , where by B(x, r) we denote the ball {y ∈ X : d(y, x) < r}. Therefore, since sup P ∈Pn diam(P ) goes to zero as n → ∞, there exists N0 ∈ N such that P (N0, x) ⊂ B(x, r) ⊂ U.
Let us write J0 = P (N0, x) and let A0 denote the element of the partition P0 such that T N 0 (J0) = A0. To get the desired result, it is enough to show (37) and (38) for the set f W = {x ∈ J0 ∩ X0 : d(T n (x), x0) < rn for infinitely many n}.
Notice that we can assume that limn→∞ rn = 0. Otherwise, there exists C > 0 such that rn > C for all n, and since µ is ergodic, from Theorem A' in [19], we deduce that for µ-almost every point in J0, d(T n (x), x0) < C for infinitely many n .
To obtain (37) we will construct, for each small ε > 0, a Cantor-like set Cε⊂ f W and we will prove using Corollary 2.1 that We construct now the Cantor-like set as follows: we start with J0 = {J0} and we denote by G −1 J 0 the composition of the N0 branchs of T −1 such that J0 = G −1 J 0 (A0). Let I(x0) = {pi} denote the sequence associated to the approximable point x0 given by Definition 6.1. Let s k = diam (P (p k , x0)), and for each s k let n(k) denotes the greatest natural number such that s k ≤ r n(k) . We denote by D the set of these indexes n(k). Since s k → 0 as k → ∞ and rn → 0 as n → ∞ by hypothesis, we have that n(k) → ∞ as k → ∞. We will write D = {di} with di < di+1 for all i.
Notice that if d ∈ D, then there exists k(d) ∈ I(x0) such that and Moreover from the property (C) of expanding maps we have that Hence, for all d large enough To construct the family J1 we first choose a natural number N1 so that d1 := N0 + N1 ∈ D, and large enough so that (16) Let SN 1 ,M 1 denote the collection of elements in PN 1 given by Proposition 3.1. We define e J1 as the family of sets G −1 J 0 (S) with S ∈ SN 1 ,M 1 , see Figure 1. Notice that by construction if e J ∈ e J1, then and since T N 0 ( e J) = S for some S ∈ SN 1 ,M 1 We remark that we will define later the family J1 by taking an appropriate subset of each one of the elements of the family e J1. From Proposition 4.2 we obtain that if e with C an absolute constant. Hence, from (14), (47), part (iii) of Theorem E and by taking N1 large enough we get that for all e J1 ∈ e J1 From (47) we also get an estimate on the size of the family e J1 Therefore, from Proposition 3.1, and the part (iii) of Theorem E we get that with C > 0 depending on P1 = A0 and P2 = P (0.x0). Now, since d1 = N0 + N1 ∈ D, by (42) and (43) there exists an integer k1 ∈ I(x0) such that and diam(P (k1, x0)) ≥ r d 1 +1 .
Moreover, from (45) we have that k1 ≤ (1+ε) 2 log β N1. Since we can take N1 as large as we want so that k1 is large enough we can get that closure (P (k1, x0)) ⊂ P (0, x0) and also by taking N1 large we have from the definition of δ λ (x0) and (49) and (45) that For all S ∈ SN 1 ,M 1 let G −1 S denote the composition of the N1 branchs of T −1 such that S = G −1 S (P (0, x0)). In each set S in SN 1 ,M 1 we take the subset L1 := G −1 S (P (k1, x0)) and we denote by L1 this family of sets. To define the family J1 we just "draw" the sets L1 in J0. More precisely, J1 is the family G −1 J 0 (L1) with L1 ∈ L1, see Figure 2.
Hence if x ∈ J ∈ J1 then T d 1 (x) ∈ P (k1, x0) ⊂ B(x0, r d 1 ), and it follows that By construction we have that for all J ∈ J1 there exists an unique e J ∈ e J1 such that J ⊂ e J, and by using the condition (A.6) and (50) we have that Also by (46), (52) and Proposition 4.2 we get that with C > 0 a constant depending on λ(P (0, x0)). And from (51) by taking N1 large, we have that Now, let us assume that we have already constructed the families e Jj, Jj and the numbers Nj and kj ∈ I(x0) for j = 1, . . . , m with the following properties: Let d1 = N0 + N1 and dj := N0 + N1 + · · · + Nj + k1 + · · · + kj−1 for j ≥ 2 .
(a) For all point x in Jj ∈ Jj (b) For all e Jj ∈ e Jj we have (b1) T d j ( e Jj) = P (0, x0) and (b2) There exists a unique Jj−1 ∈ Jj−1 so that e Jj ⊂ Jj−1 and (e) Nj is big enough so that j N1 + · · · + Nj < 1 j .
We want to mention that the hypothesis on x0 of being approximable is only required to obtain an absolute constant e c in the property (d).
Now from (56), Proposition 3.1 and again the definition of approximable points we get and this gives property (d) for j = m + 1. Observe that the constant c depends on the comparability constant between λ and µ in P1 = P (0, T km (x0)) but from the definition of approximable points we know that this constant is absolute.
As in the initial step from the definition of δ λ (x0), (58) and (54), we have that In each set S ∈ SN m+1 ,M m+1 we take the subset Lm+1 := G −1 S (P (km+1, x0)) and we call Lm+1 to this family of sets. We recall that by G −1 S we denote the composition of the Nm+1 branchs of T −1 such that S = G −1 S (P (0, x0)). To define the family Jm+1 we "draw" the family Lm+1 in each one of the sets J ∈ Jm. More precisely, for each J ∈ Jm let GJ denote the composition of the dm + km branchs of T such that GJ (J) = P (0, T km (x0)). We define now Jm+1 as Notice that by construction if J ∈ Jm+1, then Therefore the condition (c1) holds for j = m + 1. Besides, by (57), if x ∈ J ∈ Jm+1 then T d m+1 (x) ∈ P (km+1, x0) ⊂ B(x0, r d m+1 ), and therefore the condition (a) holds for j = m + 1. By construction we have that for all Jm+1 ∈ Jm+1 there exists an unique e Jm+1 ∈ e Jm+1 such that Jm+1 ⊂ e Jm+1, and by using the condition (A.6) and (50) as in the initial step we have that The estimates of λ(Jm+1)/λ( e Jm+1) of the condition (c2) follows by applying Proposition 4.2 and by using (55), (59) and (60).
We have already obtained the properties (a)-(e) for j = m + 1, and therefore we have concluded the construction of the Cantor-like set Cε. The property that for all Jm+1 ∈ Jm+1 there exists a unique Jm ∈ Jm such that closure(Jm+1) ⊂ Jm implies that Cε is not empty. And moreover Cε ⊂ X0 since by construction P (m, x) is defined for all x ∈ Cε. Hence the condition (a) implies that Cε is contained in the set W .
Next we will prove the estimate (38) for the λ-grid Hausdorff dimension of Cε. We will use the subcollections {Qm} of {Pm} given by in order to apply Proposition 2.1. Since Π is a λ-regular grid, see Definition 2.5, we only need to deal with the computation of the parameters {am} and {bm} of the subcolletions Qm. We recall that am and bm are, respectively, a lower and an upper bound for λ (P (m, x)) with x ∈ Cε.
The easiest cases correspond to m = dn and m = dn + kn, i.e. to the families e Jn and Jn. Since P (dn, x) belongs to e Jn, from property (b2) of Cε and by taking Nn large enough we have that a dn = e −Nn(hµ+3ε) and b dn = e −Nn(hµ−3ε) .
Hence from (72) we get that with C > 0 an absolute constant. In order to obtain (71) from (72) we must also say that we have taken Nn large enough so that Cn−1 ≥ −εNn.
and by property (c3) of Cε for Nn large enough we get that From (74), (75) and (76) it follows that for Nn large with C > 0 an absolute constant.
Finally, to prove (39) it is enough to show that, for all x ∈ Cε, First, notice that from the definition of the measure ν and the properties (c2) and (d) of the definition of the Cantor set Cε it follows that for all x ∈ Cε: In any case, by taking Nn+1 large enough (and therefore kn+1 also large enough) or Nn large enough (and therefore kn also large enough) we have that for Jn+1 ⊂ P (m, x) ⊆ Jn, with j = n + 1 in the case (1) and j = n in the case (2). Recall also that by the property (C) of expanding maps we have Now given a ball B = B(x, r) with center x ∈ Cε we define the natural number m = m(B) given by 2C and the family P(B) as the collection of blocks in P ∈ Pm such that P ∩ B = ∅. Let us also denote by n = n(B) the natural number such that dn + kn ≤ m < dn+1 + kn+1. It is clear that and using (84) we obtain that where j = n if dn + kn ≤ m < dn+1 and j = n + 1 if dn+1 ≤ m < dn+1 + kn+1. Notice that, by (85), it is clear that ∪ P ∈P(B) P ⊂ 2B := B(x, 2r) and since λ is a doubling measure we have that X Therefore, in each of the above cases, we have But, by the property (c2) of the Cantor set Cε and by (85) we obtain that where we have used that dn+1 Nn+1 and dn + kn dn Nn. Hence, From (86) and (87) we get (83).
Remark 6.5. If X ⊂ R and λ is Lebesgue measure, then Theorem 6.1 holds also for all approximable point x0 ∈ X + 0 . In fact, in this case we can not assure that closure(P (k1, x0)) ⊂ P (0, x0). However it is true that closure(P (k1, x0)) ⊂ P (0, x0) ∪ {x0} and from this fact we can conclude easily that where S is a countable set. Hence, Cε and T ∞ n=0 S J∈Jn closure (J) have the same Hausdorff dimensions. Also, since λ(J) = λ(closure (J)) the proof of Theorem 6.1 allows to estimate the Hausdorff dimensions of T ∞ n=0 S J∈Jn closure (J). The next result follows from the proof of Theorem 6.1. In this case the sequence of radii is constant and therefore we are estimating the set of points returning periodically to a neighbourhood of the given point x0. The proof is much more simple because the constructed Cantor-like sets have a more regular pattern. Corollary 6.1. Let (X, d, A, λ, T ) be an expanding system with finite entropy Hµ(P0) with respect to the partition P0 where µ is the ACIPM associated to the system. Let us consider the grid Π = {Pn}. Let r > 0 and let P be a block of PN 0 . Then, given ε > 0, for all point x0 ∈ X0, there exist k depending on x0 and r, and e N depending on P , x0, r and ε such that for all N ≥ e N the grid Hausdorff dimensions of the set R(P, x0, r, N ) of points x ∈ P ∩ X0 such that d(T d j (x), x0) < r for dj = N0 + k + (j − 1)(N + k) for j = 1, 2, . . .

Proof.
We have now = 0 and we can do the same construction that in Theorem 6.1 with Nj = N and kj = k for all j ≥ 1. The result for Dim Π,λ follows from Corollary 2.1 by taking αj = e −N a , βj = e −N b with a = hµ + 2ε, b = hµ − 2ε and γj a constant. Part 1 is a consequence of Proposition 2.1. The proof of Part 2 is similar to the corresponding one in the proof of Theorem 6.1. Now instead of (84) we have that ν (P (m, x)) ≤ C C n 3 λ (P (m, x)).
Remark 6.6. The lemma also holds (with the same proof) for the λ-Hausdorff dimension.
Corollary 6.2. Under the hypotheses of Theorem 6.1 we have that .
Moreover, if the grid Π is λ-regular, then and from Theorem 6.1 we obtain that .
By applying this last result to the sequence {rn/m} ∞ n=1 for any m ∈ N, we get that the lower bound in the statement follows from the above lemma. The proof of the second statement is similar.
As in the measure section we are also interested in the size of the set where L(x0) = lim sup n→∞ 1 n log 1 λ(P (tn,x 0 )) and hµ is the entropy of T with respect to µ. Moreover, for all approximable point x0 ∈ X0, the Hausdorff dimension of the set f W(U, x0, {tn}) verify: 1. If the grid Π is λ-regular, then where w = lim sup n→∞ tn n . 2. If λ is a doubling measure verifying that λ(B(x, r)) ≤ C r s for all ball B(x, r), then Remark 6.7. We recall that from Remark 6.1 and Lemma 4.4 we know that the set of approximable points such that τ λ (x0) = 0 has full λ-measure.
As in the case of radii, we have the following consequence of the proof of Theorem 6.2 when we take the sequence {tn := t} constant. We are estimating the set of points in whose code appear periodically the first t digits of the code of the point x0. The proof is similar. Corollary 6.3. Let (X, d, A, λ, T ) be an expanding system with finite entropy Hµ(P0) with respect to the partition P0 where µ is the ACIPM associated to the system. Let us consider the grid Π = {Pn}. Let t ∈ N and let P be an block of PN 0 . Then, given ε > 0 for all point x0 ∈ X0 there exist k depending on x0 and t, and e N depending on P , x0 ε and t, such that for all N ≥ e N the grid Hausdorff dimensions of the set e R(P, x0, r, N ) of points x = [ m0 m1 . . . ] ∈ P ∩ X0 such that for j = 1, 2, . . .
where C1 is an absolute constant. Moreover for all x0 ∈ X0 we have 1. If the grid Π is λ-regular then,
Proof of Theorem 6.2. We may assume that L(x0), τ λ (x0) and w are all finite, since otherwise our Hausdorff dimension estimates are obvious. Now, the proof is similar to the proof of Theorem 6.1. For each ε > 0 we construct a Cantor-like set Cε ⊂ f W (U, x0, {tn}). Recall that in the proof of Theorem 6.1 we defined an increasing sequence D of allowed indexes. Here, we define D in the following way: Let I(x0) = {pi} denote the sequence associated to x0 given by Definition 6.1. For each p k ∈ I(x0) let n(k) denote the greatest natural number such that t n(k) ≤ p k . We denote by D the set of these allowed indexes. We will write D = {di} with di < di+1 .
With this new definition of D we have that if d ∈ D, then there exists k(d) ∈ I(x0) such that t d ≤ k(d) < t d+1 and P (k(d), x0) ⊂ P (t d , x0) .
These two properties substitute to (42) and (43). For all d large enough we have that This inequality substitute to (44). With the above considerations and proceeding as in the proof of Theorem 6.1 we construct the families e Jj, Jj and the numbers Nj and kj ∈ I(x0) with the properties (b), (c1), (c3), (d) and (e). The corresponding properties (a) and (c2) are now the following ones: λ( e Jj) λ(P (kj, x0)) and λ(Jj) The rest of the proof is similar.
Remark 6.8. Using the same argument that in Remark 6.5 we get that if X ⊂ R and λ is Lebesgue measure, then Theorem 6.2 holds also for all approximable point x0 ∈ X + 0 . For points x0 ∈ X1 we have the following version of the above theorem. where w = lim sup n→∞ tn n . Moreover, if the grid Π is λ-regular, then also Proof. We may assume that w < ∞ since otherwise our estimations are trivial. The proof is similar to the proof of Theorem 6.2 but using that for d large

Upper bounds of the dimension
We will prove now some upper bounds for the λ-grid Hausdorff dimension of W(U, x0, {rn}) and f W(U, x0, {tn}) in the case that the partition P0 is finite.
Proposition 6.1. Let (X, d, A, λ, T ) be an expanding system such that the partition P0 is finite. Let µ be the ACIPM associated to the system. Let {tn} be a non decreasing sequence of positive integers and U be an open set in X with µ(U ) > 0. Then, if x0 ∈ X + 0 , we have that the grid Hausdorff dimensions of the set where L(x0) = lim infn→∞ 1 n log 1 λ(P (tn,x 0 )) , hµ = hµ(T ) is the entropy of T with respect to the measure µ and D is the cardinality of P0. Moreover, if x0 ∈ X1, then where w = lim infn→∞ 1 n tn .
Proof. We define the collections Then the collection covers the set f W(U, x0, {tn}) for all N ∈ N. Using Proposition 4.3 we get that Let us consider now the following two subcollections of P k : Then, X Since hµ ≤ Hµ(P0) ≤ log D we have that e khµ(1−τ ) ≤ D k e −kτ hµ and therefore using (88) we obtain that By part (iii) of Theorem E we know that µ(P (t k , x0)) λ(P (t k , x0)). Then for t k large enough µ(P (t k , x0)) λ(P (t k , x0)) ≤ e −k(L−ε) .
For these τ 's the τ -dimensional µ-grid Hausdorff measure of f W(U, x0, {tn}) is zero and therefore The result follows by taking ε tending to zero and using Lemma 6.1. Finally, if x0 ∈ X1, we have that L(x0) = hµ w. Remark 6.9. Even in the case that the partition P0 is not finite a slight modification of the above proof shows that To see this, notice that if we define for any ε > 0 the subcollections: P n,ε,big = {Q ∈ Pn : µ(Q) > e −n(hµ+ε) } .
then the set f W(U, x0, {tn}) ∩ X1 can be covered by the collections where F ε n.big = {T −n (P (tn, x0)) ∩ Q : Q ∈ P n,ε,big } . The proof of (89) follows now easily. Proposition 6.2. Let (X, d, A, λ, T ) be an expanding system with X ⊂ R and such that the partition P0 is finite. Let µ be the ACIPM associated to the system. Let {rn} be a non increasing sequence of positive numbers and U be an open set in X with µ(U ) > 0.
Then, for x0 ∈ X0, the Hausdorff dimensions of the set where D is the cardinality of P0, hµ = hµ(T ) is the entropy of T with respect to µ and = lim inf Notice that for n large enough B(x0, rn) ⊆ P (0, x0) and then T −n (B(x0, rn)) ∩ Q is an interval for any Q ∈ Pn. Therefore, the collection of intervals By estimating P Q∈P k µ(Q) τ as in the proof of Proposition 6.1 we get For r k small enough we have that B(x0, r k ) ⊂ P (0, x0) and then µ(B(x0, r k )) λ(B(x0, r k )) by part (iii) of Theorem E. Hence, from the definition of δ λ (x0) and Lemma 3.1, we conclude that given ε > 0, .
For these τ 's the τ -dimensional µ-Hausdorff measure of W(U, x0, {rn}) is zero and therefore The result follows by taking ε tending to zero and using Lemma 6.1. (c) f is derivable in ∪jPj and there exists σ > 0 such that |f (x)| ≥ σ for all x ∈ ∪jPj.
(e) There exists a non zero natural number m such that λ(f −m (Pi) ∩ Pj) > 0 for all i, j.
(f) There exist constants C > 0 and 0 < α ≤ 1 such that, for all x, y ∈ Pj, Markov transformations are expanding maps with parameters α and β = γ 1/n 0 , see [30], p.171, and therefore, by Theorem E, there exists a unique f -invariant probability measure µ in [0, 1] which is absolutely continuous with respect to Lebesgue measure and satisfies properties (i)-(v) in Theorem E. As a consequence of our results we obtain  Recall that, as we saw in Section 4.1, given an expanding map we have a code for almost all point x0, and more precisely for all x0 ∈ X + 0 . The following result summarizes our results about coding for Markov transformations. for infinitely many n. Then, (1) If P n λ(P (tn, x0)) = ∞, then λ( f W (x0, {tn}))) = 1. Moreover, if the partition P0 is finite or if f (P ) = [0, 1] (mod 0) for all P ∈ P0, then we have the following quantitative version: for λ-almost every x.
It is well known that we can extend the set function ν to a probability measure defined on the σ-algebra of the Borel sets of Σ . The space (Σ, σ, ν) is called a (one-sided) Bernoulli shift. We can generalize the full shift space (Σ, σ, ν) by considering the set ΣA defined by (2) pipi,j = pj , for every j = 0, 1, . . . , D − 1.
The numbers pi,j are called the transition probabilities associated to the transition matrix A and the matrix M is called a stochastic matrix. Observe that the probability vector (p0, . . . , pD−1) is an eigenvector of the matrix M .
We will explain now how to associate to (one-sided) Bernoulli shifts or (one-sided) subshift of finite type a Markov transformation: Let (Σ, σ, ν) be a (one-sided) Bernoulli shift and let λ denote the Lebesgue measure in and f equal to zero on the boundaries of the intervals Pj. It is easy to check that f is a Markov transformation and therefore an expanding map. Define now a mapping π : Σ −→ [0, 1] by Then it is not difficult to see that π is continuous and f Notice that the space of codes associated through f to the points in X0 (as we explained in Section 4.1) is precisely the set Σ0 of all sequences (i0, i1, . . . ) such that there is not k0 ∈ N such that ij = 0 for all j ≥ k0 or ij = D − 1 for all j ≥ k0, and that π is bijective from this set onto X0 = [0, 1] \ ∪ ∞ n=0 f −n (∪i∂Pi). It is also easy to check that the image measure of the product measure ν under π is precisely the Lebesgue measure in [0, 1] and that f preserves Lebesgue measure. Therefore the dynamical systems (Σ, σ, ν) and ([0, 1], f, λ) are isomorphic. We have also that the Hausdorff dimension of a Borel subset B ⊂ [0, 1] coincides que the ν-Hausdorff dimension of π −1 (B).
The subshifts of finite type whose stochastic matrix M is transitive (i.e. there exists n0 > 0 such that all entries of M n 0 are positive) can be also thought as Markov transformations: Let (ΣA, σ, ν) be a subshift of finite type with respect to the stochastic matrix M = (pi,j) and the probability vector (p0, p1, . . . , pD−1). Consider as before a partition {P0, P1, . . . , PD−1} of [0,1] in D consecutive open intervals such that λ(Pi) = pi for i = 0, 1, . . . , D − 1. We divide now each interval Pi into D consecutive open intervals Pi,j such that λ(Pi,j) = pipi,j for j = 0, 1, . . . , D−1. If pi,j = 0 we take Pi,j = ∅. Notice that by property (1) of stochastic matrices we have that λ(Pi) = P j λ(Pi,j). We define now a function f : [0, 1] −→ [0, 1] in the following way: for each Pi,j = ∅, We define also f on the points beloging to Pi ∩ ∂Pi,j in such a way that f is continuous in that points. Finally we define f to be zero on the boundaries of the intervals Pi.
When the stochastic matrix M verifies that there exists n0 such that all entries of M n 0 are positive, it is easy to check that f is a Markov transformation and therefore an expanding map with respect to the partition P0 = {Pi : i = 0, 1, . . . , D − 1}. The condition on M it is necessary only to assure property (e) of Markov transformations, see [30], Lemma 12.2. Notice also that the condition (2) in the definition of stochastic matrices means that f preserves Lebesgue measure.
Proof. First, let us observe that for subshifts of finite type X + 0 = [0, 1] because P0 is a finite partition of intervals. Also, since f preserves the Lebesgue measure λ we have that the ACIPM µ, whose existence is assured by Theorem E, coincides with λ. Then, parts (1) and (2)  The special properties of Bernoulli shifts allow us to get a better upper bound for the Hausdorff dimension of the set f W(s0, {tn}) in that case. To prove it we will use the following concentration inequality (see, for example [26]).
The above results allow us to get, for example, the following one: Corollary 7.2. Let (Σ, σ, ν) be a Bernoulli shift.
When p0 = · · · = pD−1 = 1/D we can identify the Bernoulli shift (Σ, σ, ν) with the set of D-base representations of numbers in the interval [0, 1]. The associated expanding map f is then the map f (x) = Dx (mod 1), and the measure results contained in Corollary 7.2 in this particular case, are well known (see [35]).

Gauss transformation
Let us consider now the map φ : [0, 1] −→ [0, 1] given by Here [x] denotes the integer part of x. The map φ is called the Gauss transformation and it is very close related with the theory of continued fractions. Recall that given 0 < x < 1 we can write it as If φ(x) = 0, i.e. if x / ∈ {1/n : n ∈ N} ∪ {0}, we can repeat the process with φ(x) to obtain with n1 := -.
If φ n (x) = 0 for all n, or equivalently if x is irrational, we can repeat the process for all n and associate in this way to x the infinite sequence {nj}, with nj = [1/φ j (x)] and we write x := [n0 n1 n2 . . . ] = lim Observe that if we denote by In the interval In = (1/(n + 1), 1/n), then the sequence nj is determined by the property φ j (x) ∈ In j . If x is rational the above expansion is finite (ending with n such that φ n (x) = 0. We call to the code [n0 n1 n2 . . . ] the continued fraction expansion of x. It is clear that the Gauss transformation acts on the continued fraction expansions as the left shift It is not difficult to check that the Gauss transformation φ is a Markov transformation with respect to the partition P0 = {In} and that the continued fraction expansion of x coincide with the code associated to an expanding map given in Section 4.1. It is also easy to check that φ preserves the so called Gauss measure which is defined by where λ denotes Lebesgue measure. Since this measure is obviously absolutely continuous with respect to λ, we conclude that the Gauss measure is the unique φ-invariant absolutely continuous probability whose existence is assured by Theorem E.
The next theorem is an example of the kind of statements that we can obtain when we apply our results to the Gauss transformation.  Proof. Let us observe first that now δ λ (x0) = δ λ (x0) = 1 for all x0 ∈ [0, 1] and that obviously λ and µ are comparable in [0, 1]. With this facts in mind, part (2) is a consequence of part (2) of Theorem 7.1 if x0 ∈ ∪jIj. Part (2) is also true if x0 = 1/m for some m ∈ N, since λ and µ are comparable in [0, 1] and then we do not need that B(x0, r k ) ⊂ P (0, x0) in the proof of Proposition 5.1.
Since λ and µ are comparable in [0, 1] we have that all irrational number is approximable (see Definition 6.1) and as we have just seen τ (x0) = 0 if log in = o(n), we can use Remark 2.4 and Corollary 6.2 to obtain that 1 + x dx = π 2 6 log 2 .
For continued fractions expansions there is an analogous to Corollary 7.2. However we have preferred to state the following result involving the digits appearing in the continued fraction expansion of x0. (1) λ( f W ) = 1, if X n 1 (i0 + 1) 2 · · · (it n + 1) 2 = ∞ .

Inner functions
A Blaschke product is a complex function of the type verifying the Blaschke condition P ∞ k=1 (1 − |a k |) < ∞. The function B(z) is holomorphic in the unit disk D = {z : |z| < 1} of the complex plane and it is an example of an inner function, i.e a holomorphic function f defined on D and with values in D whose radial limits f * (ξ) := lim r→1 − f (rξ) (which exists for almost every ξ by Fatou's Theorem) have modulus 1 for almost every ξ ∈ ∂D. Therefore an inner function f (z) induces a mapping f * : ∂D −→ ∂D. It is well known that any inner function can be written as where B(z) is a Blaschke product and ν is a finite positive singular measure on ∂D.
For inner functions it is well known the following result, see e.g. [39]: Theorem G (Löwner's lemma). If f : D −→ D is an inner function then f * : ∂D −→ ∂D preserves Lebesgue measure if and only if f (0) = 0.
We recall that, by the Denjoy-Wolff theorem [14], for any holomorphic function f : D −→ D which is not conjugated to a rotation, there exists a point p ∈ D, the so called Denjoy-Wolff point of f , such the iterates f n converge to p uniformly on compact subsets of D. Also, if p ∈ D then f (p) = p and if p ∈ ∂D then f * (p) = p. Hence, if f is an inner function which is not conjugated to a rotation and does not have a fixed point p ∈ D then its Denjoy-Wolff point p belongs to ∂D and f n converges to p uniformly on compact subsets of D. Bourdon, Matache and Shapiro [9] and Poggi-Corradini [38] have proved independently that if f is inner with a fixed point in p ∈ ∂D, then (f * ) n can converge to p for almost every point in ∂D. In fact, see Theorem 4.2 in [9], (f * ) n → p almost everywhere in ∂D if and only if P n (1 − |f n (0)|) < ∞. If f is inner with a fixed point in D, f preserves the harmonic measure ωp. We recall that ωp can be defined as the unique probability measure such that, for all continuous function φ : where e φ is the unique extension of φ which is continuous in D and harmonic in D. It follows that if A is an arc in ∂D, then ωp(A) is the value at the point p of the harmonic function whose radial limits take the value 1 on A and the value 0 on the exterior of A.
If f is inner with a fixed point in D, but it is not conjugated to a rotation, J. Aaronson [1] and J.H. Neuwirth [34] proved, independently, that f * is exact with respect to harmonic measure and therefore mixing and ergodic. In fact, inner functions are also ergodic with respect to α-capacity [21]. An interesting study of some dynamical properties of inner functions is contained in the works of M. Craizer. In [11] he proves that if f belongs to the Nevanlinna class, then the entropy of f * is finite and it can be calculated by the formula h(f * ) = 1 2π where (f * ) denotes the angular derivative of f . He also proves that the Rohlin invertible extension of an inner function with a fixed point in D is equivalent to a generalized Bernoulli shift, see [12]. The mixing properties of inner functions are even stronger. In this sense Ch. Pommerenke [36] has shown the following In the terminology of [19] this imply that inner functions with f (p) = p (p ∈ D) are uniformly mixing at any point of ∂D with respect to the harmonic measure ωp. In particular, we have that the correlation coefficients of characteristic functions of balls have exponential decay. As a consequence of Theorem 3 in [19], and the arguments of the proofs of Corollaries 5.2 and 5.4 we have that if ξ0 is any point in ∂D and {rn} is a non increasing sequence of positive numbers, then we have that A finite Blaschke product B (with, say, N factors) is a rational function of degree N and therefore it is a covering of order N of ∂D. As a consequence B has a fixed point in ∂D if N ≥ 3 or if N = 2 and B(0) = 0. Hence, we can choose a branch of the argument of B(e iθ ) mapping 0 on 0 and [0, 2π] onto [0, 2N π]. Also B(z) is C ∞ at the boundary ∂D of the unit disk and its derivative verifies Therefore, if B(0) = 0, we have that |B (z)| > C > 1 for all z ∈ ∂D, and the dynamic of B * on ∂D is isomorphic to the dynamic of a Markov transformation with a finite partition P0 (it has N elements) and having the Bernoulli property. Besides, since the Lebesgue measure is exact we have that the ACIPM measure of the system is precisely Lebesgue measure λ. Hence, we obtain the following improvement of statement (A): where C is a constant depending on T . Therefore the lower bound follows from the case p = 0. The equality for B(z) = z N follows from Proposition 6.2.
Theorem 7.4 is also true for the following infinite Blaschke product: since as we will see, the dynamic of B * on ∂D is isomorphic to the dynamic of a Markov transformation with a countable partition P0 and with the Bernoulli property. Notice also that B * is exact with respect to Lebesgue measure and therefore we have that the ACIPM measure is Lebesgue measure. If we denote B(e 2πit ) = e 2πiS(t) then S (t) = |B (e 2πit )| > C > 1. Moreover, it follows from Phragmén-Lindelöf Theorem that the image of S(t) is (−∞, ∞) and so we can define the intervals

Expanding endomorphisms
Let M be a compact Riemannian manifold. A C 1 map f : M −→ M is an expanding endomorphism if there exists a natural number n ≥ 1 and constants C > 0 and β > 1 such that (Dxf n )u > C β n u , for all x ∈ M, u ∈ TxM .
A C 1 expanding endomorphism of a compact connected Riemannian manifold M whose derivative Dxf is a Hölder continuous function of x is an expanding map with respect to Lebesgue measure λ and a finite Markov partition P0, see [30], p.171. Therefore, the unique f -invariant probability measure whose existence is assured by Theorem E is comparable to λ in the whole M . Our results also apply for this dynamical system.