Approximations for Gibbs states of arbitrary Holder potentials on hyperbolic folded sets

In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $\Lambda$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary H\"{o}lder potentials, with the help of weighted sums of atomic measures on preimage sets of high order. Our endomorphism may have also stable directions on $\Lambda$, thus it is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (they depend on the whole past and may intersect each other both inside and outside $\Lambda$), and moreover the number of preimages remaining in $\Lambda$ may vary; also Markov partitions do not always exist on $\Lambda$. Our convergence results apply also to Anosov endomorphisms, in particular to Anosov maps on infranilmanifolds.


Introduction
Gibbs states (equilibrium measures) of Hölder potentials for smooth maps appear naturally in statistical mechanics (for example [2], [15], [16], etc). In order to study equilibrium measures for smooth diffeomorphisms, one can use the specification property in order to find plenty of periodic points, which in turn can be used to approximate equilibrium measures. This method was employed successfully by Bowen in [2] (see also [3]). Also in [2] it is studied the weighted distribution of preimages for one-sided shifts of finite type (which are examples of non-invertible maps).
The setting that we work with in this paper is different from both the hyperbolic diffeomorphism case (see [2]), as well as from the expanding case (see [13]). We assume that f : M → M is a smooth map (endomorphism), not necessarily a diffeomorphism, and that Λ is a basic set for f so that f is hyperbolic on Λ, but not necessarily expanding. For such non-expanding endomorphisms there are no Markov partitions in general, so it is not possible to code the system using Markov partitions like in the diffeomorphism case. Also the presence of stable directions on Λ makes the local inverse iterates of small balls to grow exponentially (up to a certain order). For instance in [1] it was introduced a family of horseshoes with self intersections, and it was proved that there are open sets of parameters which give non-injectivity of the map on the respective basic set. In the noninvertible case, if f is hyperbolic on Λ, we do not have a foliation structure for the local unstable manifolds; the local unstable manifolds depend now on the whole prehistories (see [14]). This folding of the phase space is a major difference from the diffeomorphism case, since we are forced to work onΛ which is not a manifold. The local unstable manifolds may intersect each other and through any given point there may pass infinitely many local unstable manifolds. Moreover the number of f -preimages belonging to Λ may vary from point to point, so the map is not necessarily constant-to-1; the set Λ is not necessarily totally invariant. The local unstable manifolds depend Holder on their respective prehistories in the canonical metric onΛ ( [8]). The existence of several n-preimages in Λ for any point x ∈ Λ means that we can have n-preimages y ∈ f −n (x)∩Λ of x where the consecutive sum S n φ(y) := φ(y) + . . . + φ(f n−1 y) is well behaved, but also other n-preimages z ∈ f −n (x) ∩ Λ where S n φ(z) is badly behaved.
In the case of smooth non-invertible expanding maps the situation is difficult, and the problem of finding the weighted distributions of preimages was solved by Ruelle in [13]; in that situation it was important that the local inverse iterates contract uniformly on small balls.
In our present non-invertible non-expanding setting, we will describe the weighted distributions of preimages by studying the intersections between different tubular neighbourhoods of the (many) different local unstable manifolds, and by the use of specification and the expression for equilibrium measure with the help of periodic points. This will imply also the use of some finer properties of the lifting of invariant measures to the natural extension and the comparison between the different types of behaviors of weighted sums of atomic measures on various prehistories, with respect to Gibbs states. The main results of the paper are in Theorem 1 and its Corollaries: Theorem. Let f : M → M be a smooth map (say C 2 ) on a smooth Riemannian manifold M , so that f is hyperbolic and finite-to-one on a basic set Λ; assume also that the critical set C f of f does not intersect Λ. Let also φ a Holder continous potential on Λ and µ φ be the equilibrium measure of φ on Λ. Then In Corollary 1 we obtain an approximation for the equilibrium measure µ φ , i.e the weak-⋆ convergence of a sequence of weighted atomic probabilities of the above type towards µ φ .
Corollary. In the same setting as in Theorem 1, for any Holder potential φ with equilibrium measure µ φ , it follows that there exists a subset E ⊂ Λ, with µ φ (E) = 1 and an infinite subsequence (n k ) k such that for any z ∈ E we have the weak-⋆ convergence of measures: In particular, if µ 0 is the measure of maximal entropy, it follows that for µ 0 -almost all points x ∈ Λ, We remark that, since µ φ is positive on any open set (as any open set contains some small Bowen ball and one can apply Proposition 1), there exists a dense set in Λ of points x for which we have the above weak convergence of weighted atomic measures generated by x, towards µ φ . Therefore in a physical non-reversible system, if one knows the past trajectories of such a generic point x up to some high level n, then one can approximate the Gibbs state µ φ as above.
One may also compare this result with the usual (forward) SRB measure in the case of diffeomorphisms (see for example [2], [15], [16]) or endomorphisms ( [4], [12], [11]); in fact whenever µ φ is equivalent to the Lebesgue measure, like in the case of toral endomorphisms and φ ≡ 0, we obtain an inverse SRB result. Our setting and methods are however different due to the lack of an inverse function, the fact that unstable manifolds depend on whole prehistories (not just base points), and also to the fact that the number of preimages is not necessarily constant on Λ. We will apply the L 1 Ergodic Theorem of von Neumann on the natural extensionΛ for the lifted measure, together with combinatorial arguments in order to estimate the measure µ φ on the intersections between different tubular unstable sets. Then we will estimate the equilibrium measure on the different parts of the consecutive preimage sets in Λ, by carefully studying the prehistories from the point of view of convergence properties of certain weighted sums of Dirac measures along them. In our Theorem let us notice that we average over all n-preimages of points, so we do not consider only one prehistory. This simultaneous consideration of all n-preimages is what makes the proof difficult. It cannot be obtained just by applying Birkhoff Ergodic Theorem to different prehistories, since the speeds of convergence may be different over the uncountable collection of prehistories.
Among the examples of smooth endomorphisms on folded basic sets, let us mention the horseshoes with self-intersections from [1], the hyperbolic skew products with overlaps in their fibers from [10], or dynamical systems generating from certain non-reversible statistical physics models (see [15]). Examples may be obtained also from non-degenerate holomorphic maps on complex projective spaces (for instance [9], [7]).
Another application of Theorem 1 will be in Corollary 2, where it will be applied to Anosov endomorphisms, in order to give the distribution of consecutive preimage sets, with respect to different equilibrium measures. A classical example of Anosov endomorphism is given by a toral endomorphism f A : T m → T m , m ≥ 2, where f A is the map induced on the m-dimensional torus by a matrix A with integer coefficients and detA = 0. Then any point in T m has exactly |detA| f Apreimages in T m , as can be seen since the f A image of the unit square is a parallelogram with area (volume) equal to |detA|, whose corners have integer coordinates ( [17]). If A has all its eigenvalues of absolute values different from 1, then f A is a hyperbolic endomorphism and the above Theorem will apply. Since the equilibrium measure of any constant function is the Haar measure ( [17]), we obtain the asymptotic distribution of the local inverse iterates toward an inverse SRB measure in this case. A generalization of this class of examples is given by smooth perturbations of hyperbolic toral endomorphisms on T m , m ≥ 2. They will be again constant-to-1 and we will be able to apply our main Theorem, to obtain the weighted distribution of preimages with respect to equilibrium measures of Holder potentials.
Remark: Even on algebraic-type manifolds, like infranilmanifolds, the situation of Anosov endomorphisms is very different from that of Anosov diffeomorphisms. Indeed in the case of infranilmanifolds ( [6], [18]), Franks and separately Manning showed that any Anosov diffeomorphism can be "linearized", i.e it is topologically conjugate to some hyperbolic automorphism. Also, Gromov ( [6]) showed that if f is an expanding map on a compact manifold, then f is topologically conjugate to some expanding endomorphism on some infranilmanifold. However this is not the case for Anosov endomorphisms. As was proved in [18], if M is an infranilmanifold then there exists a C 1 dense subset U in the set of "true" Anosov endomorphisms on M (i.e those endomorphisms which are not Anosov diffeomorphisms nor expanding maps), such that every f ∈ U is not shift equivalent (hence also not topologically equivalent) to any hyperbolic infranilmanifold endomorphism. In particular this applies to tori T m , m ≥ 2, which are natural examples of infranilmanifolds. For such Anosov endomorphisms which are neither diffeomorphisms nor expanding, one cannot apply results similar to the ones from those two previous cases. However we can apply Theorem 1 to get the distribution of preimage sets with respect to equilibrium measures.

Distributions of consecutive preimages on basic sets
First let us establish some notations. The next definition is parallel to that of basic set from [3] (see also [7]).
Definition 1. Let f : M → M a smooth map (say C 2 ) defined on the smooth manifold M . We will say that a compact f -invariant set Λ is a basic set for f if there exists a neighbourhood U of Λ such that Λ = ∩ n∈Z f n (U ), and if f is transitive on Λ. As we work here with non-invertible maps, such sets will also be called folded basic sets. Definition 2. The natural extension of the dynamical system (f, Λ) is the dynamical system (f ,Λ), whereΛ : will be called an n-prehistory of x.
Let us mention that if µ is an f -invariant probability measure on Λ, then there exists a unique probabilityf -invariant measureμ onΛ such that π * (μ) = µ. It can be seen that µ is ergodic if and only ifμ is ergodic onΛ. Also the topological pressure of φ (denoted by P f (φ) to emphasize dependence on f ) is equal to the topological pressure of φ • π, namely Pf (φ • π); and µ is an equilibrium measure for φ : Λ → R if and only ifμ is an equilibrium measure for φ • π.
The concept of hyperbolicity on Λ makes sense for non-invertible maps (i.e endomorphisms), but now the unstable tangent subspaces and the local unstable manifolds depend on whole prehistories, not only on the base points (for exp. [14]). In this hyperbolic setting we will denote by E s x , W s r (x) the stable tangent subspace, respectively the local stable manifold at x (for x ∈ Λ); and by E û x , W u r (x) the unstable tangent subspace, respectively the local unstable manifold corresponding to the prehistoryx (forx ∈Λ). Also by Df s (x) we shall denote the stable derivative Df x | E s x and by Df u (x) the unstable derivative Df x | E û x . In [8] we studied the Holder dependence of local unstable manifolds with respect to the prehistories and proved a Bowen type formula, giving the unstable dimension as being the zero t u of the pressure of the unstable potential Φ u (x) := − log |Df u (x)|,x ∈Λ. We proved also that given a measurable partition ofΛ subordinated to the unstable manifolds, the equilibrium measure of Φ u has conditional measures which are geometric of exponent t u .
If f : M → M is a smooth map which is hyperbolic on a basic set Λ, letx = (x, x −1 , . . . , x −n , . . .) ∈ Λ, n ≥ 1, ε > 0 small. Then we call an (n, ε)-tubular unstable neighbourhood (or tubular unstable set) the set The notion of tubular unstable set can be extended to those y ∈ M which have an n-preimage y −n ∈ M with the above property, but since we will work in this paper only with measures supported on Λ, we preferred to give the definition restricted to Λ. It is important to keep in mind that tubular unstable sets corresponding to two different prehistories of the same point x ∈ Λ may not be the same; still they intersect in a set containing x.
It is well-known that any f -invariant measure µ on Λ can be lifted to a uniquef -invariant measureμ onΛ such that π * (μ) = µ. It will be important to see exactly how to calculate the measureμ of an arbitrary closed set fromΛ, in terms of the µ-measures of sets in Λ.
Lemma 1. Let f : Λ → Λ be a continuous map on a compact metric space Λ, and µ an f -invariant borelian probability measure on Λ. Letμ be the uniquef -invariant probability measure onΛ with the property that π * (μ) = µ. Then for an arbitrary closed setÊ ⊂Λ, we have that Proof. The arbitrary closed setÊ is not necessarily of the form π −1 (E) for some E borelian set in Λ. Let us denoteÊ n :=f −nÊ , n ≥ 1; thenμ(Ê n ) =μ(Ê) sinceμ isf -invariant. Let alsô F n := π −1 (π(Ê n )), n ≥ 1. We will prove that We have clearlyÊ ⊂f n (F n ), n ≥ 0. Let now a prehistoryẑ ∈ ∩ n≥0f nF n ; then ifẑ = (z, z −1 , . . . , z −n , . . .), we obtain that z −n ∈ πÊ n , ∀n ≥ 0, henceẑ ∈Ê sinceÊ is assumed closed. Thus we showed the equalityÊ = ∩ n≥0f n (F n ). Now let us notice that the above intersection is decreasing, sincê f n+1F n+1 ⊂f nF n , n ≥ 0, since for a prehistory fromf n+1F n+1 the (n + 1)-th entry is from πÊ n+1 and the n-th entry is in πÊ n , whereas the (n + 1)-entry of a prehistory fromf nF n can be any preimage of a point from πÊ n . Since the above intersection is decreasing, we get We used that π * μ = µ and thatμ isf -invariant onΛ. Therefore we obtain thatμ(Ê) = lim For a basic set Λ for a smooth map f we will denote by In general in this paper we will be interested only in the preimages belonging to Λ. We will denote also by S n φ(y) for a point y ∈ Λ, the consecutive sum S n φ(y) := φ(y) + . . . + φ(f n−1 y), n ≥ 1. Define then for n ≥ 1 and x ∈ Λ, the probability measure: The µ x n , n ≥ 1 are probability measures and thus from the weak compactness of the unit ball in the space of measures, we obtain that any sequence of such measures (for x ∈ Λ given) contains a convergent subsequence; the limit of such a sequence (µ x n k ) k≥1 is then an f -invariant probability measure on Λ. We will show that such limit measures have in fact an important thermodynamical property, namely they are equilibrium measures (i.e maximize in the Variational Principle, [17]).
We shall use in the sequel equilibrium measures for endomorphisms; the existence of these measures for non-invertible maps, and estimates on Bowen balls for these measures are similar to the corresponding properties for diffeomorphisms (see the proof of Proposition 1 below). We denote the Bowen ball B n (y, ε) := {z ∈ Λ, d(f i y, f i z) < ε, i = 0, . . . , n − 1}, for y ∈ Λ, n ≥ 1, ε > 0. Proposition 1. Let Λ be a hyperbolic basic set for a smooth endomorphism f : M → M , and φ be a Holder continuous function on Λ. Then there exists a unique equilibrium measure µ φ for φ on Λ which has the following properties: a) for any ε > 0 there exist positive constants A ε , B ε and an integer n 0 ≥ 1 such that for any b) By working eventually with a finite iteration of f , we have Proof. a) We work in the natural extensionΛ with the expansive homeomorphismf :Λ →Λ. The existence of a unique equilibrium measure for the Holder potential φ • π with respect to the expansive homeomorphismf :Λ →Λ follows from the standard theory for homeomorphisms on compact metric spaces (see for example [2], [3]); let us denote this equilibrium measure byμ φ . Then, given anf -invariant probability measureμ onΛ, there exists a unique f -invariant measure µ on Λ such that π * μ = µ. If we take the measureμ φ instead ofμ, we will obtain a measure µ φ . It is easy to show that hμ(f ) = h µ (f ) and that Pf (φ • π) = P f (φ), ∀φ ∈ C(Λ, R). Thus it follows that µ is an equilibrium measure for φ if and only if its uniquef -invariant liftingμ is an equilibrium measure for φ • π onΛ. Thus µ φ := π * μφ is the unique equilibrium measure for φ on Λ. Now we see that there exists a k = k(ε) ≥ 1 such thatf k (π −1 B n (y, ε)) ⊂ B n−k (f kŷ , 2ε) ⊂Λ, for any y ∈ Λ. On the other hand for anyŷ ∈Λ, we have π(B n (ŷ, ε)) ⊂ B n (y, ε). These two inclusions and thê f -invariance ofμ φ , together with the estimates for theμ φ -measure of the Bowen balls inΛ (from [3]) imply that there exist positive constants A ε , B ε (depending on ε > 0 and on φ) such that: So the estimates for Bowen balls are true also for endomorphisms.
b) The iterate of f may be needed in order to have topological mixing (needed to guarantee specification, [3]). However without loss of generality we may assume that f is topologically mixing on Λ. If x is a periodic point for f | Λ , say f m (x) = x, then we obtain a periodic point forf , namely the prehistoryx = (x, f m−1 (x), . . . , f (x), x, . . . , f (x), x, . . .) ∈Λ. Conversely, ifx is a periodic point forf , then x is a periodic point (of the same period) for f . Similarly as for diffeomorphisms we prove that if f is hyperbolic then f satisfies specification. Then specification is used to show the convergence of the weighted sums of Dirac measures concentrated at periodic points towards µ φ , in the same way as in [3].
Let us prove now a Lemma giving a relationship between the measures of different parts of the preimage of some set of positive measure. Lemma 2. In the above setting, let a Holder potential φ : Λ → R and its unique equilibrium measure µ φ . Let us consider ε > 0, k disjoint Bowen balls B m (y 1 , ε), . . . , B m (y k , ε) and a borelian set ε). Then there exists a positive constant C ε independent of m, y 1 , . . . , y m such that Proof. We shall denote the equilibrium measure µ φ by µ to simplify notation, and will work with the restriction of f to Λ (without saying this explicitly every time). Similarly as in [3] or in [17], since the borelian sets with boundaries of measure zero form a sufficient collection, we may assume that each of the sets A i , A j have boundaries of µ-measure zero.
From construction f m (A i ) = f m (A j ), i, j = 1, . . . , k. But µ can be considered as the limit of the sequence of measuresμ Let us consider now a periodic point x ∈ Fix(f n ) ∩ A i ; by definition of A i , it follows that f m (x) ∈ A i , so there exists a point y ∈ A j such that f m (y) = f m (x). Of course the point y does not have to be periodic necessarily. But we can use Specification Property ( [3], [2]) on the hyperbolic compact locally maximal set Λ. Indeed if ε > 0 is fixed, then there exists a constant M ε > 0 such that for all n > M ε , there is a z ∈ Fix(f n ) s.t z ε-shadows the (n − M ε )-orbit of y.
Let now V be an arbitrary neighbourhood of the set A j . Then if n is large enough and ε > 0 fixed, it follows that z ∈ V . Let us also take two points x,x ∈ Fix(f n ) ∩ A i and assume the same periodic point z ∈ V ∩ Fix(f n ) corresponds to both through the previous procedure. This means that the (n − M ε − m)-orbit of f m z ε-shadows the (n − M ε − m)-orbit of f m x and also the (n − M ε − m)-orbit of f mx . Hence the (n − M ε − m)-orbit of f m x 2ε-shadows the (n − M ε − m)-orbit of f mx . But recall that we chose x,x ∈ A i ⊂ B m (y i , ε), hencex ∈ B n−Mε (x, 2ε). Now we can split the set B n−Mε (x, 2ε) in at most N ε smaller Bowen ball of type B n (ζ, 2ε). In each of these (n, 2ε)-Bowen balls we may have at most one fixed point for f n . This holds since fixed points for f n are solutions to the equation f n ξ = ξ and, on tangent spaces we have that Df n − Id is a linear map without eigenvalues of absolute value 1. Thus if d(f i ξ, f i ζ) < 2ε, i = 0, . . . , n and if ε is small enough, it follows that we can apply the Inverse Function Theorem at each step. Therefore there will exist only one fixed point for f n in the Bowen ball B n (ζ, 2ε). So there may exist at most N ε periodic points from Fix(f n ) ∩ Λ which have the same point z ∈ V attached to them by the above procedure. In conclusion, to each point x ∈ Fix(f n ) ∩ A i , there corresponds at most finitely many points z ∈ V obtained by specification from the above procedure; their number is smaller than N ε . Let us notice also that if x,x have the same point z ∈ V attached to them, then as seen beforex ∈ B n−Mε (x, 2ε) and thus, from the Holder continuity of φ: for some positive constantC ε depending on φ (but independent of n, x). This can be used then in the estimate forμ n (A i ), according to (2). Now we use the fact that if z ∈ B n−Mε (y, ε), then f m (z) ∈ B n−Mε−m (f m x, ε). From the Holder continuity of φ and the fact that x ∈ A i ⊂ B m (y i , ε), it follows that there exists a constantC ε (we denote it the same as before, without loss of generality) satisfying: for n > n(ε, m). This, together with (2) and the fact that there are at most N ε points x ∈ Fix(f n ) which have the same attached z ∈ V ∩ Fix(f n ), imply that there exists a constant C ε > 0 such that where we recall that A i ⊂ B m (y i , ε), A j ⊂ B m (y j , ε). But since ∂A i , ∂A j have µ-measure zero, we obtain: e Smφ(y i ) e Smφ(y j ) , i, j = 1, . . . , k Now V was chosen arbitrarily as a neighbourhood of A j , thus Similarly we prove also the other inequality.
Theorem 1. Let f : M → M be a smooth (say C 2 ) map on a Riemannian manifold M , which is hyperbolic and finite-to-one on a basic set Λ so that C f ∩Λ = ∅. Assume that φ is a Holder continuous potential on Λ and that µ φ is the equilibrium measure of φ on Λ. Then with the notation from (1), Proof. We make the convention that all the preimages that we work with are in Λ. So we shall write f −n x for f −n Λ x, n ≥ 1, x ∈ Λ. If φ is a Holder continuous function on Λ if follows from Proposition 1 that there exists a unique equilibrium measure µ φ for φ, and µ φ is the push-forward of the equilibrium measureμ φ of φ • π onΛ. For simplicity of notation, we shall denote the measure µ φ by µ, with φ being fixed. This measure is ergodic as being an equilibrium measure.
Let us fix now a continuous test function g : Λ → R. From von Neumann's L 1 Ergodic Theorem applied to the homeomorphismf −1 :Λ →Λ and the potential g • π , we know that where the prehistoryx = (x, Hence for an arbitrary small η > 0, we have from (5) that: So for any ε ′ > 0, ε ′ = ε ′ (η) << η, there exists n(η) ≥ 1 so that if n > n(η) then µ(x, |Σ n (g, x −n )| ≥ η) < ε ′ But now {x ∈Λ, |Σ n (g, x −n )| ≥ η} is a closed set inΛ, thus we can apply Lemma 1 to prove that: if n is large enough ( without loss of generality we can take n > n(η)). Now let us consider a small ε > 0 with ε < ε(η) << η such that ω(3ε) < η, where ω(r) denotes in general the maximal oscillation of g on a ball of radius r > 0. Let us take also a maximal set of mutually disjoint n-Bowen balls B n (y, ε) in Λ; denote the set of such y by F n . Thus {B n (y, ε), y ∈ F n } is our maximal set. If z / ∈ F n , then from the definition, B n (z, ε) must intersect some Bowen ball B n (y, ε), y ∈ F n . Thus B n (z, ε) ⊂ B n (y, 3ε). Let us notice also that if w ∈ B n (z, 3ε) then |Σ n (g, w)| ≤ |Σ n (g, z)| + ω(3ε) In the sequel we will split different subsets of F n in two disjoint subsets R n , G n , with R n ⊂ {x ∈ Λ, |Σ n (g, x)| ≥ 2η} and G n ⊂ {x ∈ Λ, |Σ n (g, x)| < 2η}. Intuitively R n consists of the "bad" n-preimages (corresponding to g, η) and G n are the "good" n-preimages.
Recall now that we denoted f −n Λ x := Λ ∩ f −n x, n ≥ 1, x ∈ Λ, and that Λ is not necessarily totally invariant. Consider then e Snφ(z) · |Σ n (g, y)|, x ∈ Λ, n ≥ 1 Denote also by Λ(n, η) := {x ∈ Λ, s. t x has at least one n − preimage x −n with Σ n (g, The problem is that the unstable manifolds may depend on the whole prehistories, thus if we take z, w ∈ f −n x, then f n B n (w, ε) and f n B n (z, ε) may be different unstable tubular sets; these unstable tubular sets intersect each other in a (possibly smaller) set containing x. By taking n-preimages for all points in F n and then the corresponding tubular unstable sets as above, we shall obtain a collection of such tubular unstable sets which intersect each other in different smaller pieces, denoted generically by D. We have to estimate Λ I n (g, x)dµ(x), which is a sum of integrals on sets of type D. The problematic terms in this sum are those of type with |Σ n (g, y)| ≥ 2η, but so that at the same time there exist also good n-preimages for every point x ∈ D . In other words x ∈ Λ(n, η), but x has also good n-preimages. The set of points where all n-preimages from Λ are bad, is easier to measure, by using Lemma 1. In the integral Λ I n (g, x)dµ(x) we have to deal both with the measures of subsets D, and with the n-preimages y of points x ∈ D, namely with the quantities Σ n (g, y)|.
For the measures of subsets D we shall use the f -invariance of µ, namely µ(D) = µ(f −n (D)). Then we will use the estimate on the measure of the set of bad preimages from Lemma 1, coupled with the existent control on the good preimages given by |Σ n (g, y)| < η. For the rest of the preimages, the idea is to control the sum of the measures of these inverse iterates by the measure of the set of bad preimages.
First let us notice that if x ∈ Λ and y, z ∈ f −1 x ∩ Λ, y = z, then since the critical set C f does not intersect Λ, it must exist a positive constant ε 0 so that d(y, z) > ε 0 . Hence if similarly for some n ≥ 1, we take two distinct n-preimages y, z ∈ Λ of x (f n y = f n z = x, y = z), then we cannot have y, z ∈ B n (w, 3ε) for any w ∈ Λ, if ε << ε 0 . We know also that the Bowen balls B n (y, 3ε) cover Λ when y ranges in F n , and B n (y i , ε) ∩ B n (y j , ε) = ∅ for any two different points y i , y j from F n .
Let us take then a small 0 < β < 1 such that η < β; to fix ideas we will consider β = 3η. If x ∈ Λ, denote by R n (x) the set of n-preimages y ∈ Λ of x with |Σ n (g, y)| > 2η (in fact R n (x) depends on the η as well, but we do not record this here in order to simplify notation). Let us denote now Now if ||g|| := sup y∈Λ |g(y)|, it follows from definition that for any y ∈ Λ, we have |Σ n (g, y)| ≤ 2||g||; thus for a point x ∈ D n (β, η): We will now restrict to the complement of D n (β, η) in Λ, so we work with points x ∈ Λ for which y∈Rn(x) e Snφ(y) y∈f −n x e Snφ(y) ≥ β. From Proposition 1 we know that for any z ∈ Λ, n ≥ 1, A ε · e Snφ(z)−nP (φ) ≤ µ(B n (z, ε)) ≤ B ε · e Snφ(z)−nP (φ) . Therefore we have Obviously from the definition of F n we can place any n-preimage of x in a distinct Bowen ball of type B n (z, 3ε), for some z ∈ F n . From Proposition 1 it follows that the µ-measure of a Bowen ball of type B n (z, ε) is comparable to the µ-measure of the Bowen ball B n (z, 3ε) (by comparable we mean that their quotient is bounded below and above by a positive constant independent of z, n). The idea is to split now the set Λ \ D n (β, η) into subsets A such that points from A have the same number of n-preimages. Let us assume that d is the maximum number of 1-preimages that a point may have in Λ. Then any point from Λ has at most d n n-preimages in Λ. We will use now tubular unstable sets T n (x, 3ε) obtained from different Bowen balls centered at the points of F n . For an integer ℓ ≥ 2 let us consider the sets of type where B j := B n (y j , 3ε) for some points y j ∈ F n , and where we assume no repetitions among the respective Bowen balls. This means actually that we do not repeat a Bowen ball B j in the above intersection; if y, z are different n-preimages of the same point x and ε is small enough, then we cannot have y, z in the same B n (ζ, 3ε). Each point in such a set has exactly ℓ n-preimages, one in each of the Bowen balls B i 1 , . . . , B i ℓ .
We will take an arbitrary subset S ∈ H(ℓ, n, ε), . From the f -invariance of µ, it follows that µ(S) = µ(f −n S) = µ(f −n S ∩ B i 1 ) + . . . + µ(f −n S ∩ B i ℓ ). We have that the Bowen balls B is , s = 1, . . . , ℓ are mutually disjoint as they contain different n-preimages of the same point (we know that for small ε one cannot have two different n-preimages of the same point, belonging to the same B n (y, 3ε), y ∈ Λ). Let us denote by S n (i 1 ) := f −n S ∩B i 1 , . . . , S n (i k ) := f −n S ∩ B i k . So µ(S) = µ(S n (i 1 ) + . . . + µ(S n (i k )) We assume that µ(S n (i 1 )) > 0, otherwise we can take a different S n (j). But now from Lemma 2 and the fact that S ⊂ Λ \ D n (β, η), it follows that Suppose that T is another disjoint set from some H(p, n, ε), so that T ⊂ f n (B i 1 )∩f n (B j 2 )∩. . .∩ f n (B jp ) and let T n (i 1 ), T n (j 2 ), . . . , T n (j p ) be the corresponding parts of f −n T belonging respectively to B i 1 , B j 2 , . . . , B jp . So the sets S n (i 1 ), . . . , S n (i ℓ ), T n (i 1 ), T n (j 2 ), . . . , T n (j p ) are mutually disjoint borelian subsets. We assumed that S and T have both n-preimages in B i 1 (for example). If they have n-preimages in completely different Bowen balls, then the situation will be simpler, since there will be no repetitions below in (12).
Then similarly as in (13), by employing the control on the total measure of bad n-preimages from (6), we can conclude that for n > n(η): Therefore we obtain that µ z n k → k µ φ , z ∈ E, i.e we have weak convergence of the measures µ z n k , k ≥ 1, on a set of z having full µ φ -measure in Λ.
The Theorem applies to Anosov endomorphisms in particular.
Corollary 2. Assume that f : M → M is an Anosov endomorphism without critical points on a Riemannian manifold. Let also φ a Holder continuous potential on M and µ φ the equilibrium measure of φ. Then We can compare these results to the usual SRB measure for the endomorphism f , defined as a measure µ having the property that for any measurable partition η ofM subordinate to the lifts of the local unstable manifolds, and forμ almost allx ∈M , the projection of the conditional measure ofμ, namely π * (μ η x ) is absolutely continuous with respect to the induced Lebesgue measure on W û x ([4], [12]). In [12] it is shown that µ satisfies the SRB property for the Anosov endomorphism f if and only if 1 Corollary 3. Let f : M → M be an Anosov endomorphism, φ : Λ → R a Holder potential and assume that the equilibrium measure µ φ is absolutely continuous with respect to the Lebesgue measure on M . Then the measure µ φ with this property is unique, it is an SRB measure and it also satisfies an inverse SRB condition in the sense that there exists a set E of full Lebesgue measure in M and a sequence (n k ) k such that µ z n k → k µ φ , z ∈ E.
Proof. The proof follows immediately from Theorem 1, Corollary 2 and from the results of [4] and [12]. The potential φ for which µ φ is SRB, can be taken in fact to be the unstable potential.
A classical example of Anosov endomorphism without critical points is a toral hyperbolic endomorphism f A : T m → T m , associated to an m×m integer valued matrix A whose eigenvalues λ i all have absolute values different from 1, and whose determinant det(A) is not necessarily 1 in absolute value. Each point from T m has exactly |detA| preimages in T m . If we consider the potential φ ≡ 0, then the equilibrium measure of φ is the Haar measure ω on T m which is also the measure of maximal entropy (its entropy is equal to log |λ i |, where each eigenvalue is taken with its multiplicity). In [17] it was proved the asymptotic distribution of periodic points towards ω; here we prove the convergence towards ω, which is also the unique measure of maximal entropy, of the measures µ x n , n corresponding to the potential φ ≡ 0, for ω-almost all points x ∈ T m . We thus obtain the existence of an inverse SRB measure in this case.
Moreover Theorem 1 applies also to smooth (say C 2 ) perturbations f A,ε , of hyperbolic toral endomorphisms f A . Indeed they will also be hyperbolic on the m-dimensional torus T m and the basic set considered is the whole T m . Also the non-invertible map f A,ε remains |det(A)|-to-1 on T m . We thus obtain the weighted distribution (with respect to a Holder potential φ on T m ) of preimage sets of f A,ε , with respect to the equilibrium measure µ φ of φ, for perturbations of hyperbolic toral endomorphisms.
Anosov endomorphisms on infranilmanifolds represent a generalization of toral linear endomorphisms (see the Remark at the end of Section 1). Let us notice that our Theorem 1 applies to Anosov endomorphisms on infranilmanifolds which are not topologically conjugate to Anosov diffeomorphisms nor to expanding maps. Thus, besides Theorem 1, one cannot apply any of the previously known results for the distributions of preimages from the case of diffeomorphisms ( [2]), or expanding endomorphisms ( [13]).
Theorem 1 applies also to hyperbolic basic sets of saddle type for endomorphisms which are not necessarily Anosov, like the class of examples from [10], namely skew products with overlaps in their fibers F : X × V → X × V, F (x, y) = (f (x), h(x, y)), where f : X → X is an expanding map on a compact metric space, while h(x, ·) : V → V (denoted also by h x ) is a contraction on an open convex set V ⊂ R m ; h x is assumed to depend continuously on x ∈ X. The basic set is in this case given by where h n z := h f n−1 z •. . .•h z , n ≥ 1, z ∈ X. In [10] we studied the conditional measures of equilibrium states induced on fibers and their relation to the stable dimension of fibers. So from Theorem 1 we obtain the weighted distributions of preimages of the non-invertible map F over Λ, with respect to equilibrium measures of Holder potentials.
We can collect the above remarks in the following: Corollary 4. a) The conclusions of Corollary 2 hold in particular for toral hyperbolic endomorphisms and for smooth perturbations of these. b) The conclusions of Theorem 1 hold for the basic sets of hyperbolic skew products with overlaps in their fibers from [10], as well as for the attractors of the noninvertible horseshoes from [1].