TOPOLOGICAL ENTROPY OF LEVEL SETS OF EMPIRICAL MEASURES FOR NON-UNIFORMLY EXPANDING MAPS

. In this article we obtain a variational principle for saturated sets for maps with some non-uniform speciﬁcation properties. More precisely, we prove that the topological entropy of saturated sets coincides with the smallest measure theoretical entropy among the invariant measures in the accumulation set. Using this fact we provide lower bounds for the topological entropy of the irregular set and the level sets in the multifractal analysis of Birkhoﬀ averages for continuous observables. The topological entropy estimates use as tool a non-uniform speciﬁcation property on topologically large sets, which we prove to hold for open classes of non-uniformly expanding maps. In particular we prove some multifractal analysis results for C 1 -open classes of non-uniformly expanding local diﬀeomorphisms and Viana maps [1, 33]. ,

1. Introduction. The study of the thermodynamic formalism and multifractal analysis for maps with some hyperbolicity has drawn the attention of many researchers from the theoretical physics and mathematics communities in the last decades. The general concept of multifractal analysis, that can be traced back to Besicovitch, is to decompose the phase space in subsets of points which have a similar dynamical behavior and to describe the size of each of such subsets from the geometrical or topological viewpoint. We refer the reader to [20,22] and references therein. A first natural problem is the multifractal analysis of Birkhoff averages. Given a continuous map T of a compact metric space X and an observable φ : X → R, it is natural to decompose where X α = {x ∈ M : lim n→∞ 1 n S n φ(x) = α} are level sets of convergence for Birkhoff averages and the irregular set I φ (T ) is the set of points for which the Birkhoff averages for φ does not converge. The description of these level sets arise in the analysis of level sets in several important quantities in dynamics. For instance, in the special case of a C 1 interval map f , the level sets on the multifractal 5408 XUETING TIAN AND PAULO VARANDAS decomposition associated to the Birkhoff averages of the potential φ = log |f | coincides with the set of points with the same Lyapunov exponent.
From the measure theoretical viewpoint, Birkhoff's ergodic theorem guarantees that the irregular set has zero measure for every invariant measure. Nevertheless, irregular sets may have full topological entropy and full Hausdorff dimension [22]. A multifractal analysis program has been carried out successfully to deal e.g. with the Lyapunov spectrum, self-similar measures and local entropies [26,20,21,4,30,9,14,15,5,28,12] in contexts of uniform and non-uniform hyperbolicity. Some methods to describe the topological entropy of level sets include the use of the differentiability and convexity of the pressure function, the existence of large deviations rate functions or the use of a specification property. Despite the fact that it provides stronger results, the description of the level sets of Birkhoff averages using the strict convexity of the pressure function is often possible only for observables that are at least Hölder continuous.
A major question in multifractal analysis is to describe the topological entropy or Hausdorff dimension of the so called saturated sets. A saturated set in X is the subset of points x ∈ X whose accumulation points V T (x), in the weak * topology, of the empirical measures in the space of T -invariant probability measures coincides with a prescribed subset of invariant probability measures. Saturated sets can be used to describe convergence properties of Birkhoff averages with respect to every continuous observable. On the one hand, by weak * convergence, V T (x) is a singleton if and only if the Birkhoff averages of every continuous observable are convergent at x. On the other hand, if the accumulation set V T (x) contains invariant measures µ 1 , µ 2 that distinguish a continuous potential φ (i.e. so that φ dµ 1 = φ dµ 2 ) then x is a Birkhoff irregular point associated to φ. Therefore, the global understanding of saturated sets in the space of probability measures allows to derive as a consequence, important results on the multifractal analysis of Birkhoff averages for continuous observables. One of the key difficulties to estimate the topological entropy of saturated sets consists of the fact that, since these enclose information on the Birkhoff averages of all continuous observables, no information on the strict convexity and differentiability of the pressure function can be used, even in a uniformly hyperbolic context. These difficulties were overcomed by Takens and Verbistki [26] and by Pfister and Sullivan [23] that used some notions similar to specification, to characterize the topological pressure of saturated sets in the case of maps with some hyperbolicity, including the Maneville-Pomeau map and β-shifts. Some of the difficulties that arise in the use of the previous methods to obtain multifractal analysis results for multidimensional non-uniformly expanding maps are that the pressure function is much harder to describe and that specification is most likely to fail in the absence of uniform hyperbolicity (see e.g. [18,25] and references therein).
In the present paper we contribute to the description of saturated sets for nonuniformly expanding maps. Although expanding and hyperbolic measures satisfy a non-uniform specification property [19,31] (in rough terms, any finite pieces of orbits of generic points can be shadowed by a true orbit of the dynamics and that the time lag between pieces of orbits grow sublinearly on the size of the pieces of orbits) these notions have been established for generic points of ergodic and invariant measures. For that reason, such notions are not a suitable tool to describe multifractal analysis since 'shadowable points' consist of Birkhoff regular with respect to every continuous observable. For that reason, we provide a criterium for non-uniformly expanding maps to admit a topologically large set of points with (topological) non-uniform specification properties (cf. Theorem 2.3). In particular, this provides a criterium for creating Birkhoff irregular points using orbits of points which are generic for different ergodic measures, something that was not possible using [19,31]. This consists of a method different from the improved shadowing lemma developed by C. Liang, G. Liao, W. Sun and X. Tian [17] in the context of non-uniformly hyperbolic diffeomorphisms. Our main result (Theorem 2.1) is that, under (topological) non-uniform specification properties, the topological entropy of the set of points whose empirical measures accumulates on two ergodic measures is bounded below by the minimum entropy among both measures. Using this, we provide a description of multifractal analysis of Birkhoff averages for continuous potentials for both irregular set and level sets (Corollaries 1, 2 and 3). We use our main results to study the multifractal analysis of Birkhoff averages for multidimensional non-uniformly expanding maps including Viana maps (we refer the reader to Section 2.3 for precise statements).
2. Statement of the main results. For any K ⊂ M T (X) it is natural to consider both sets It follows from the work of Pfister and Sullivan [23] that for any non-empty, compact Here we provide an extension of this result that makes use of some notions of nonuniform specification (we refer the reader to Subsection 4.3 for the definitions). Our main result here is as follows.
(2) if T has strong non-uniform specification on ∆, then where Some comments are in order. If T has strong non-uniform specification on ∆ then it is expected that if K ⊂ M T (X) is a non-empty, compact and convex set then This is coherent with the second item of Theorem 2.1 since, in the case that }} the affine property of the measure theoretical entropy implies that min{h tµ1+(1−t)µ2 (T ) : t ∈ [0, 1]} = min{h µ1 (T ), h µ2 (T )}. Although we do not pursue that here, it is also expected our method to produce estimates on the topological pressure of saturated sets.
The classical approach to prove that the irregular set of continuous observables that are not cohomologous to a constant has full topological entropy uses specification to create points whose Birkhoff averages oscillate between space averages with respect to two ergodic measures µ 1 , µ 2 . In this article we describe the set of points whose empirical measures have a prescribed accumulation set. In rough terms, given ergodic measures µ 1 and µ 2 , the proof of Theorem 2.1 follows the strategy of Pfister and Sullivan [23] to bound the entropy of the set of points whose empirical measures accumulate on a convex set K by the entropy of a Cantor sets whose topological entropy is bounded below by the smallest entropy of the measures in K.
The construction of such a Cantor set uses the non-uniform specification property on some set that contains the basin of different ergodic and expanding measures. In particular, we can use these results to provide some results on the multifractal analysis of open classes of multidimensional non-uniformly expanding maps derived from expanding and Viana maps [33,32].

Multifractal analysis of Birkhoff averages. Given a continuous function
For convenience, we refer to R φ (T ) as the regular set with respect to φ (or simply φ-regular set). The φ-irregular set is defined as I φ (T ) = X \ R φ (T ). These two sets describe different asymptotic behavior under the observation of the given continuous function φ. Clearly, if φ is cohomologous to a constant, meaning that φ = u•T −u+c for some c ∈ R and u : X → R continuous, then the Birkhoff averages converge at every point and the irregular set is empty. The next results show that if the dynamics satisfies some specification property then the topological entropy of the irregular set is bounded below by the entropy of the measures supported in a set with non-uniform specification. The notions of non-uniform specification are defined in Subsection 4.3 below.

2.2.1.
Maps with strong non-uniform specification. For any continuous function φ : X → R and any a ∈ R the φ-regular set can be further decomposed in level sets. Indeed, given a ∈ R let and observe that R φ (T ) can be written as the following disjoint union Theorem 2.1 can be used to derive a variational relation for Birkhoff level sets. We need some preliminary definitions. Define X ∆ := {x ∈ X| V T (x) ∩ M T (∆) = ∅}, and the regular sets R ∆,φ (T ) := R φ (T ) ∩ X ∆ and the level sets R ∆,φ,a (T ) := R φ,a (T ) ∩ X ∆ . The set X ∆ arises naturally in the proof of the variational principle for non-compact invariant sets. Dually, consider the set X * ∆ := {x ∈ X| V T (x) ⊆ M T (∆)} formed by points whose accumulation of empirical measures lie in M T (∆) and let R * ∆,φ (T ) and R * ∆,φ,a (T ) be defined similarly as before (taking X * ∆ instead of X ∆ ). Clearly, X * ∆ ⊆ X ∆ ⊆ X and the equalities hold e.g. if M T (∆) = M T (X). The relative Birkhoff irregular sets for φ are defined similarly by I ∆,φ (T ) := I φ (T ) ∩ X ∆ and I * ∆,φ (T ) := I φ (T ) ∩ X * ∆ . Corollary 1. Let T : X → X be a continuous map of a compact metric space X, let ∆ ⊆ X and let φ : X → R be a continuous function. If T has strong non-uniform specification on ∆, then (1) Given any real number a ∈ R, h top (T, R ∆,φ,a (T )) = h top (T, R * ∆,φ,a (T )) = sup{h ρ (T )| ρ ∈ M T (∆) and φ dρ = a}.

Maps with non-uniform specification.
In what follows we also consider dynamics where do not require the strong non-uniform specification. The first result concerns the irregular set of continuous observables. Corollary 2. Let T : X → X be a continuous map of a compact metric space X, let ∆ ⊆ X and φ : X → R be a continuous function. Suppose that T has non-uniform specification on ∆. If then In particular, if T admits a maximal entropy measure supported on ∆ then I φ (T ) carries full entropy.
Some comments are in order. It is well known that if T satisfies the specification property then the following are equivalent: (i) inf µ∈M T φ dµ < sup µ∈M T φ dµ, (ii) the Birkhoff averages of φ are not uniformly convergent to a constant, and (iii) the Birkhoff irregular set I φ (T ) is non-empty. Moreover, the Livsic theorem for mixing uniformly expanding maps assures that the later conditions hold if and only if φ is not cohomologous to a constant, or equivalently, there are expanding periodic points p 1 and p 2 so that φ dδ O(p1) < φ dδ O(p2) . In many applications of our results to non-uniformly expanding maps, the set ∆ ⊆ X of points where nonuniform specification holds will be taken as the set of points whose sequence of hyperbolic times is non-lacunar, a notion that is intimately related to the integrability of the first hyperbolic time map (cf. Subsection 7.1).
Returning to the multifractal analysis, if, instead of its strong version, the dynamical system satisfies the non-uniform specification property then one could not estimate the topological entropy of level sets. Nevertheless, given a continuous potential φ, a ∈ R and σ > 0 one can estimate the size of the following sets obtained as approximations of level sets R φ,a,σ (T ) :

More precisely
Corollary 3. Let T : X → X be a continuous map of a compact metric space X, let ∆ ⊆ X and φ : X → R be a continuous function. Given any real number a ∈ R, if T has non-uniform specification on ∆, then 1. for any σ > 0, 2.3. Non-uniformly expanding maps. Throughout this subsection let T be a C 1 -map on X which behaves like a power of the distance to a critical/singular region The choice of these constants is important for the construction of hyperbolic times (see e.g. [1]). Definition 2.2. Assume that η ∈ M T (X). We say that (T, η) is non-uniformly expanding if there exists σ > 1 such that η-almost every x satisfies lim sup and the slow recurrence condition: for every ε > 0 there exists δ > 0 such that, for η-almost every point x ∈ X, The non-uniform hyperbolicity property for an ergodic measure µ implies on the exponential non-uniform specification property on its basin of attraction B(µ) [19,31]. We are interested in extending this fact to sets that contains the basin of several invariant measures. This motivates the following definition.
We say that T is non-uniformly expanding if where, for any σ > 1 and δ > 0, M σ,δ ⊂ M T (X) denotes the space of T -invariant probability measures η that have a non-lacunar sequence of (σ, δ)-hyperbolic times. It is known that if the first hyperbolic time is integrable then the sequence of hyperbolic times is non-lacunar (see e.g. [32]). We refer the reader to Subsection 7.1 for the definitions. The following result provides a criterium for a non-uniformly expanding map to satisfy the (strong) exponential non-uniform specification property.
Theorem 2.3. Let X be a compact Riemannian manifold and let T : X → X be a topologically exact C 1 -endomorphism. If M = ∅ then T satisfies the (strong) exponential non-uniform specification property on a set ∆ ⊂ X such that η(∆) = 1 for every η ∈ M. More precisely, for any points x 1 , x 2 , . . . , x k in ∆ there exists δ > 0 and σ > 1 (depending on the points) so that the following property holds: for any n 1 , . . . , n k ≥ 1 there exists y ∈ X so that We observe that the dependence of the constants δ, σ on the points can be dropped out, for instance in the case that the set ∆ consists of generic points for a finite number of ergodic and expanding measures (cf. Remark 1). 3. Some examples. It is clear both the non-uniform and strong non-uniform specification properties hold for uniformly hyperbolic diffeomorphisms and expanding maps. In particular, our results extend some well known results on the multifractal analysis in the uniformly hyperbolic setting. Moreover, our results can be applied for the one-dimensional Maneville-Pomeau transformations and Benedicks-Carleson quadratic maps. In what follows we focus on providing some new results on the multifractal analysis of multidimensional non-uniformly expanding maps.
3.1. Local diffeomorphisms with non-uniform expansion. The first class of examples we shall consider are multidimensional local diffeomorphisms obtained by local bifurcation of expanding maps, considered originally by Alves, Bonatti and Viana [1].
Let T 0 be an expanding map in T n and take a periodic point p ∈ T n for T 0 . Let U be a C 1 -open set of local diffeomorphisms so that every T ∈ U is a C 1 -local diffeomorphism obtained from T 0 by a bifurcation in a small neighborhood V of p in such a way that: Although the original expanding maps satisfy the specification property it is expected that most local diffeomorphisms in U should not satisfy the specification property (we refer the reader to [25] for such a discussion in the case of partially hyperbolic diffeomorphisms). The coexistence of expanding and contracting behavior constitutes an obstruction to a global description of the multifractal description of the dynamics. We describe some applications in two situations: (a) Equilibrium states for potentials with low variation. If φ : T n → R is a Hölder continuous potential so that then there exists a unique equilibrium state µ φ for T with respect to φ, it is fully supported and it is an expanding measure with an integrable first hyperbolic time map ( [32]). In particular T has a unique maximal entropy measure. Theorem 2.3 implies that there exists a dense set ∆ ⊂ T n where T satisfies the strong (exponential) non-uniform specification property. There exists t 0 > 1 so that the potential tφ also satisfies the variation condition (10) for every t ∈ [−t 0 , t 0 ]. To the best of our knowledge the following was the only result known on the multifractal analysis of these class of dynamics: the pressure function [9,Proposition 7.2]). Our main result, Theorem A (2) implies that Moreover, if ψ : T n → R is any continuous observable so that ψ dµ 1 = ψ dµ 2 for some expanding measures µ 1 = µ 2 then it follows from Corollary 2 that the irregular set carries full topological entropy, that is, h top (T, I ψ (T )) = h top (T ).
(b) SRB measures. It follows from [1] that T has a unique (ergodic) probability measure µ 1 absolutely continuous with respect to Lebesgue, with density bounded away from zero and infinity and integrable first hyperbolic time map. Moreover, µ 1 is an equilibrium state for the potential φ = − log | det Df | (that might not satisfy the low variation condition) and is an expanding measure. If ∆ is as above and then Corollary B implies that h top (T, I φ (T )) ≥ h µ1 (T ).

3.2.
Non-uniformly expanding endomorphisms with critical region. We also prove that the C 3 -robust class of multidimensional non-uniformly expanding maps with singularities known as Viana maps also satisfies the non-uniform specification property on a dense set.
In [33], Viana introduced a robust class of multidimensional non-uniformly hyperbolic maps with singularities. More precisely, these are obtained as C 3 small perturbations of the skew product T of the cylinder S 1 × I given by where d ≥ 16 is an integer, a is a Misiurewicz parameter for the quadratic family, and α is small. These maps admit a unique SRB measure µ (it is absolutely continuous with respect to m = Leb, has only positive exponents and dµ/dm ∈ L p (m) where p = d/(d − 1)) and are strong topologically mixing on the attractor Λ = ∩ n≥0 T n (S 1 × I): for every open set W there exists n ≥ 1 such that T n (W ) = Λ. See [33,2] for more details. Since T carries full supported expanding measures then there exists a dense set ∆ ⊂ Λ of points where the strong non-uniformly expanding property holds. We observe that there are infinitely many expanding measures, including periodic Dirac masses at repelling periodic points. Moreover, it follows from [24] that T has a unique expanding map that is a maximal entropy measure and, consequently, for any continuous observable φ so that the function µ → φ dµ over all expanding measures is not constant.
and (ii) if one considers a collection {Y i } +∞ i=1 of subsets of X then

XUETING TIAN AND PAULO VARANDAS
Let M(X) denote the space of all Borel probability measures supported on X. Let ξ = {V i | i = 1, 2, · · · , k} be a finite partition of measurable sets of X. The entropy of ν ∈ M(X) with respect to ξ is Given Λ ⊂ Z we write T ∨n ξ := k∈Λ T −k ξ. The entropy of ν ∈ M T (X) with respect to ξ is h(T, ν, ξ) := lim  x, ε)).
Brin and Katok [8] proved that for any µ ∈ M e T (X), h µ (T, x) = h µ (T, x) for µ a.e. x ∈ X, and h µ (T, x) dµ(x) = h µ (T ). So for µ ∈ M T (X), We also need to recall Katok's definition of metric entropy (see [16]). It is defined for ergodic measures and equivalent to the classical one. Let Z ⊂ X. A set S is (n, ε)-separated for Z if S ⊂ Z and d n (x, y) > ε for any x, y ∈ S and x = y. A set S ⊂ Z if (n, ε)-spanning for Z if for any x ∈ Z, there exists y ∈ S such that d n (x, y) ≤ ε. Let µ ∈ M e T (X). For ε > 0, ρ ∈ (0, 1) and l ≥ 1, let N µ l (ε, ρ) be the minimal number of ε−balls B l (x, ε) in the d l −metric, which cover a set Z ⊆ X of measure at least 1 − ρ. Define Then from [16] we know 4.2. Invariant measures. Let {ϕ j } j∈N be a dense subset of C(X, R) and set ϕ i = max{|ϕ i (x)| : x ∈ X}. The expression defines a metric on M(X) which is compatible with the weak * topology (see [34]). Observe that ρ(ξ, τ ) ≤ 2 for any ξ, τ ∈ M(X).
It is well known that the natural projection x → δ x is continuous and, if we define the push-forward operator T f on M(X) by T f µ(A) = µ(T −1 (A)), we can identify (X, f ) with T f restricted to the set of Dirac measures (these are conjugate). Therefore, assume without loss of generality that the metric d is inherited from ρ, defined by d(x, y) = ρ(δ x , δ y ). The following facts are simple consequences of (15) and the fact that ρ(δ x , δ y ) = d(x, y): 1. If 0 ≤ k < n < m then ρ(E m (x), E n (T k (x))) ≤ 2 m (m − n + k); 2. Given ε > 0 and p ∈ N, for every y ∈ B p (x, ε) we have ρ(E p (y), E p (x)) < ε; 3. Given ε > 0 and p, q ∈ N satisfying p ≤ q ≤ (1 + ε/2)p, for every y ∈ B p (x, ε) we have ρ(E q (y), E p (x)) < 2ε.

4.3.
Some forms of specification. In the last years some different versions of non-uniform specification have been introduced, many of them unrelated. Here we define the notions that will be used throughout this paper. For any n ∈ N, the d n -distance between x, y ∈ X is defined as Let x ∈ X, n ≥ 1 and ε > 0. The dynamical ball B n (x, ε) is defined as the set Definition 4.2. We say that T satisfies the non-uniform specification property on the set ∆ ⊆ X, if there exists δ > 0 such that for every x ∈ ∆, every n ≥ 1 and every 0 < ε < δ there exists a positive integer p(x, n, ε) ≥ 1 so that lim ε→0 lim sup n→∞ 1 n p(x, n, ε) = 0 and the following holds: given points x 1 , · · · , x k in ∆ and positive integers n 1 , · · · , n k , if p i ≥ p(x i , n i , ε) then there exists z ∈ X that ε-shadows the orbits of each x i during n i iterates with a time lag of p i in between T ni (x i ) and x i + 1, that is, In the case that T is a topologically exact C 1 -local diffeomorphism and µ is a T -invariant and ergodic probability measure with infinitely many hyperbolic times then f satisfies the non-uniform specification property on the set B(µ), where B(µ) = x ∈ M : 1 n n−1 j=0 δ T j (x) → µ in the weak * -topology stands for the ergodic basin of attraction of µ (see [31]). Here we also consider two similar-flavored notions with a (summable) control of the distances during the 'shadowing'. for every 0 < ε < δ and the shadowing sizes during the non-uniform specification process satisfy This condition is weaker than the following one, which requires the shadowing distances to be exponentially small and that holds for broad classes of non-uniformly expanding maps (recall Theorem 2.3).
Definition 4.4. We say that T satisfies the exponential non-uniform specification property with respect to the exponent λ > 0 on the set ∆ ⊆ X, if non-uniform specification property holds on ∆, lim sup n→∞ 1 n p(x, n, ε) = 0 for every 0 < ε < δ and the shadowing sizes in the non-uniform specification can be taken exponentially small, i.e., In the previous definitions the term 'non-uniform' refers to the time lag between the shadowing of the pieces while the terms 'strong' and 'exponential' refer to sharper approximations of the distances involved in the shadowing. Specification properties have been proved to be strongly related to hyperbolicity. The original specification property was first introduced by Bowen in [7] (see also [10]) and requires the function p(x, n, ε) to depend just on ε. Thus, in comparison with nonuniform specification properties, the time required to be able to shadow the finite pieces of orbits does not depend on their size. Similarly, exponential specification is stronger than exponential non-uniform specification in the same sense (see e.g. [29]). Indeed the following relations hold: mixing hyperbolic basic pieces ⇓ exponential specification ⇒ exponential non-uniform specification ⇓ ⇓ strong non-uniform specification ⇓ specification ⇒ non-uniform specification 5. Proof of Theorem 2.1.
Take l large enough such that µ 1 (∆ θ ε,l ) > 1 − ρ 0 , µ 2 (∆ θ ε,l ) > 1 − ρ 0 . Let g : N → {1, 2} be given by g(k) = (k + 1)(mod 2) + 1. Choose a strictly decreasing sequence δ k → 0 and a strictly increasing sequence L k → ∞ so that the set J k := {x ∈ ∆ θ ε,l : ρ(E n (x), µ g(k) ) < δ k for all n ≥ L k } satisfies µ g(k) (J k ) > 1 − ρ 0 for all k ≥ 1. This is a simple consequence of the fact that the ergodic basin of attraction of µ i is a full µ i -measure set. Let E k n be a maximal (n, 4ε)-separated set for J k . Notice that a maximal (n, 4ε)-separated set for J k is also a (n, 4ε)-spanning set for J k and thus E k n is also a (n, 4ε)-spanning set for J k . Since µ g(k) (J k ) > 1 − ρ 0 , then by (18) lim inf Take n k be a strictly increasing sequence such that n k ≥ L k , 1 n k log #E k n k > h µ g(k) (T ) − 3γ and n k n k+1 → 0. We may assume in addition that each n k θ is always integer. Let S k := E k n k , µ k = µ g(k) , m k = n k θ. From above construction, we have Step 2. Construction of the fractal F. The purpose here is to construct a fractal F with topological entropy bounded below by h * and in such a way that the accumulation points of empirical measures associated to points of F remain ε-close to the line segment K = {tµ 1 + (1 − t)µ 2 : t ∈ [0, 1]}. Let us choose a sequence with N 0 = 0 and N k increasing to ∞ sufficiently quickly so that For any (x 1 , . . . , x k ) ∈ S N1 1 × . . . × S N k k , by non-uniform specification property, we have For every k ≥ 1, consider the set F k defined as Note that each set of F k is non-empty and compact. Moreover, since F k+1 ⊆ F k for every k ≥ 1 then F := ∞ k=1 F k is compact and non-empty. Lemma 5.1. For any p ∈ F, d Haus (V T (p), K) < . In other words, F ⊆ QG µ1,µ2 ( ).
Proof. The proof is standard.
By Lemma 4.1 items (1) and (2), the distance between empirical measures of each of the x i s can be bounded as: Using Lemma 4.1 (1) one more time we deduce that Then lim sup k→∞ ρ(E t k (p), µ k ) ≤ . For other subsequences the strategy of the proof follows well known ideas by constructing an appropriate convex sum of the measures µ 1 and µ 2 . Here we omit the details.
Step 3. Construction of a special sequence of measures ω k . We must first undertake an intermediate construction.
. . x k ). If T k is the set of all points constructed in this way we compute its cardinality below.
are distinct then any z 1 := z(x) and z 2 := z(y) are different points. In particular #T k = #S N1 Proof. Since x = y, there exists i, j such that x i j = y i j . We have Let ω be an accumulation point of (ω k ) k≥1 in the weak * topology.
Proof. For any fixed l and every p ≥ 0, notice that ω l+p (F l+p ) = 1 and F l+p ⊆ F l .
The strategy of the proof is to use (cf. equation (13)) that and to provide lower bounds on the lower entropy of probability measures supported on the fractal F . Let B := B n (q, ε) be an arbitrary dynamical ball with B ∩ F = ∅.
We suppose that j ≥ 1 (and leave the details of the easier case j = 0 to the reader). Now we start to estimate the number of points in the set B ∩ T k+p . Proof. Firstly let us assume that p = 1 and prove that ω k+1 (B) ≤ (#T k ) −1 (#S k+1 ) −j . We need an upper bound estimate for the number of points which can be in the set T k+1 ∩ B. If ω k+1 (B) = 0 we are done. Thus we may assume throughout that ω k+1 (B) > 0 and, in particular, k+1 and some point z = z(x, x k+1 ) ∈ T k+1 ∩ B. Fix x and x k+1 as above and consider the set C x;x1,...,xj = {z(x, y 1 , . . . , y N k+1 ) ∈ T k+1 : y 1 = x 1 , . . . , y j = x j }.
This corresponds roughly to the 'cylinder' formed by all possible choices of points that shadow the coordinate elements that determine the point x and the first j components of the point x k+1 .
Proof of the claim. Take z = z(y, y k+1 ) ∈ T k+1 ∩ B for some y and y k+1 . We prove that x l = y l for l ∈ {1, 2, . . . , j} (since the proof that x = y is completely similar we shall omit it). Since z, z ∈ B = B n (q, ε) then Assume by contradiction that there exists 1 ≤ l ≤ j so that y l = x l , and let a l = t k + (l − 1)(n k+1 + m k+1 ). Recall that d n k+1 (x l , y l ) > 4ε. Using then we obtain that which contradicts (24). This proves the claim.
We proceed with the proof of the lemma. By the claim we have that ν k+1 (B) ≤ #C x;x1,...,xj = (#S k+1 ) N k+1 −j and, consequently, This proves the lemma in the case that p = 1. In the case that p > 1, similar estimates as above yield that Dividing the latter by #T k+p , it follows that ω k+p (B) ≤ (#T k ) −1 (#S k+1 ) −j proving the lemma.
To finalize the proof of Theorem 2.1 (1) we are left to give the estimate on the topological entropy of the set QG µ1,µ2 ( ). Using (20), (17) and m i = n i θ, we have Thus This proves the theorem in this first case that t k ≤ n < t k+1 − m k .
Proof of Theorem 2.1 (2). From (2), we know that Thus in order to prove (5), we need to prove We assume without loss of generality that the measures µ 1 , µ 2 are ergodic. In general, since one can use the ergodic decomposition of invariant measures exactly as in [23, Lemma 6.2] we shall omit the details here. Let h * = min{h µ1 (T ), h µ2 (T )}. We need to show that for any γ > 0, Then (5) holds. Now we fix γ > 0 and start to prove (28).
Step 1. Choice of separated sets. Fix a number ρ 0 ∈ (0, 1). For the measures µ 1 and µ 2 , by Katok's definition of metric entropy, take small number ε > 0 such that for i = 1, 2, Fix a rational number θ 0 > 0 such that Take a strictly decreasing sequence of positive rational numbers θ n ↓ 0 with θ 1 < θ 0 . Let g : N → {1, 2} be given by g(k) = (k + 1)(mod 2) + 1. Choose a strictly decreasing sequence δ k → 0 and a strictly increasing sequence L k → ∞ so that the set p(x, n, ε) n < θ k and ρ(E n (x)µ g(k) ) < δ k for all n ≥ L k } of points whose empirical measures are δ k -close to µ g(k) satisfies µ g(k) (J k ) > 1 − ρ 0 for all k ≥ 1. This is a simple consequence of the fact that the ergodic basin of attraction of µ i is a full µ i -measure set. Let E k n be a maximal (n, 4ε)-separated set for J k . Notice that a maximal (n, 4ε)-separated set for J k is also a (n, 4ε)-spanning set for J k and thus E k n is also a (n, 4ε)-spanning set for J k . Since µ g(k) (J k ) > 1 − ρ 0 , then by (29) lim inf Take (n k ) k a strictly increasing sequence of positive integers such that n k ≥ L k , 1 n k log #E k n k > h µ g(k) (T ) − 3γ and n k n k+1 → 0. We may assume in addition that each n k θ k is always integer. Let , m k = n k θ k . From above construction, we have Step 2. Construction of the fractal F. We mimic the construction of F as in the proof of Theorem 2.1 (1), by considering , T i (y)) < ε}, instead of the sets defined in (22). Here the non-emptyness of B(x 1 , . . . x k ) follows from the strong non-uniform specification property. After the construction of F , we need to replace Lemma 5.1 by following lemma.
Proof. The proof is analogous to the one of Lemma 5.1, replacing (23) by the following estimate Here the part 1 , E n k (T (j−1)(n k +m k ) p k )) ≤ 1 n k ε follows from strong non-uniform specification: Using Lemma 4.1 (1) we deduce that Then lim sup k→∞ ρ(E t k (p), µ k ) = 0. For other subsequences the strategy of the proof follows well known ideas by constructing an appropriate convex sum of the measures µ 1 and µ 2 . Here we omit the details.
Then one can follow Step 3 and 4 same as the proof of Theorem 2.1 (1) to end the proof of Theorem 2.1 (2), just in Step 4 replacing QG µ1,µ2 ( ) by G µ1,µ2 , replacing m i = n i θ by m i = n i θ i , and replacing (25) by ≥ exp (h * −3γ) 1+θ0 n .
6. Multifractal analysis. In this section we prove the corollaries of Theorem 2.1 to the multifractal analysis of Birkhoff averages for continuous observables. For any t ≥ 0, define the set Q(t) := {x ∈ M : ∃µ ∈ V T (x) so that h µ (T ) ≤ t}, which can be empty. From [6, Theorem 2] we know: Proof of Corollary 1. Let T : X → X be a continuous map of a compact metric space X and assume that T satisfies the strong non-uniform specification property on ∆ ⊆ X. Let φ : X → R be a continuous function. In order to prove (1), fix a ∈ R let D = {µ ∈ M T (∆)| φdµ = a} and take t = sup{h µ (T ) : µ ∈ D}. As R ∆,φ,a ⊆ Q(t) then it follows from (34) that h top (T, R ∆,φ,a (T )) ≤ t. Thus, to prove the Corollary 1(1) it is enough to prove that h top (T, R * ∆,φ,a (T )) ≥ h µ (T ) for every µ ∈ D. Notice that G µ,µ ⊆ R * ∆,φ,a (T ). Together with relation (5) in Theorem 2.1 this implies that h top (T, R * ∆,φ,a (T )) ≥ h top (T, G µ,µ ) ≥ h µ (T ) and proves (1).
Since τ was chosen arbitrary this completes the proof of the corollary.
Proof of Corollary 3. Let T : X → X be a continuous map of a compact metric space X and ∆ ⊆ X so that T has non-uniform specification on ∆. Let φ : X → R be a continuous function. Since the item (2) is a direct consequence of item (1) we only need to prove the later. Given a ∈ R and σ > 0 let D σ = {µ ∈ M T (∆)| φ dµ ∈ (a − σ, a + σ)}.
The proof is now complete.