Contribution to the ergodic theory of robustly transitive maps

In this article we intend to contribute in the understanding of the ergodic properties of the set RT of robustly transitive local diffeomorphisms on a compact manifold M without boundary. We prove that there exists a C^1 residual subset R_0 of RT such that any f in R_0 has a residual subset of M with dense pre-orbits. Moreover, C^1 generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbit, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results hold for an important class of robustly transitive maps considered in [Lizana-Pujals'12].


Introduction and Statement of the Main Results
In the last two decades many advances have been made to the study of robustly transitive diffeomorphisms, whose geometric properties are by now very well understood. In fact, it follows from Bonatti, Díaz, Pujals [1] that robustly transitive diffeomorphisms exhibit a weak form of hyperbolicity, namely, dominated splitting. The ergodic aspects of robustly transitive diffeomorphisms have called the attention of many authors recently. For instance, let us refer to the construction of SRB measures (see [3]) and maximal entropy measures (see [2]) for the class of DA-maps introduced by Mañé, and more recently, [12] proved intrinsic ergodicity(unique entropy maximizing measure) for partially hyperbolic diffeomorphisms homotopic to a hyperbolic one on the 3-torus.
The theory is much more incomplete in the non-invertible setting, in which case the study of robust transitivity has received far less attention. Since the negative iterates of an endomorphism are not easy to describe, the dynamics can be very hard to explain. Nevertheless, the first important contributions in this respect were given recently in [6,7], where it is shown that there are robustly transitive local diffeomorphisms without any dominated splitting, and some necessary and sufficient conditions for robust transitivity of local diffeomorphisms are given. Our purpose here is to give a contribution to the better understanding of robustly transitive local diffeomorphisms and to give first results on their ergodic theory.
Let M be a compact Riemannian manifold. We say that an endomorphism f : M → M is transitive if there exists x ∈ M such that its forward orbit by f , O + f (x) = {f n (x)} n≥0 , is dense in M. In the theory of differentiable dynamical systems, it is an important issue to know when a special feature is exhibited in all nearby systems (with respect to some topology), that is, a dynamical property is robust under perturbation. In particular, a map f is C r robustly transitive (r ≥ 1) Date: January 28, 2014.
if there exists a C r open neighborhood U(f ) of f such that every g ∈ U(f ) is transitive.
We focus our attention to local diffeomorphisms, that is endomorphisms without critical points. Let us denote by RT the set of C 1 local diffeomorphisms on M that are robustly transitive. It is not hard to check that any endomorphism that admits a dense subset of points with dense pre-orbits is transitive. In our first result we address a sort of converse to the previous assertion for generic maps. More precisely, Theorem A. There exists a C 1 residual subset R 0 ⊂ RT such that for any f ∈ R 0 the following conditions hold: (1) Periodic points are dense in M .
(3) There exists a residual subset D ⊂ M of points such that for any In the remaining, our goal is to show that robustly transitive local diffeomorphisms are interesting from the ergodic theory point of view. For that discussion let us recall some necessary definitions. Given a compact forward invariant set Λ, we say that f | Λ has no splitting in a C 1 robust way if there exists a C 1 open neighborhood U(f ) of f so that for all g ∈ U(f ) the tangent space T Λ M does not admit non-trivial invariant subbundles. We denote by RT * ⊂ RT the open subset of C 1 robustly transitive local diffeomorphisms that have no splitting in a C 1 robust way. Let SRT * ⊂ RT * denote the set of local diffeomorphisms so that all points have dense pre-orbits. It is easy to see that SRT * has non-empty interior and we consider in SRT * the induced topology from RT * . Moreover, an ergodic f -invariant probability measure µ is expanding if all the Lyapunov exponents are positive. Finally, we say that (f, µ) has exponential decay of correlations if there are constants K, α > 0 and λ ∈ (0, 1) such that for all ψ ∈ C α (M, R), ϕ ∈ L 1 (µ) and n ∈ N: Our next result illustrates that generically robustly transitive maps exhibit many ergodic measures with interesting dynamical meaning.
Theorem B. There exists a C 1 residual subset R 1 ⊂ SRT * such that for any f ∈ R 1 there are uncountable many f -invariant, ergodic and expanding measures with full support and exponential decay of correlations.
Some comments are in order. We note that, in opposition to the case of diffeomorphisms discussed in [1], there are open sets of local diffeomorphisms with robust non-existence of splitting (see Section 4 below). Moreover, there are open subsets of robustly transitive local diffemorphisms that do not admit invariant expanding measures, e.g. hyperbolic endomorphisms on T n . Clearly, these examples admit some non-trivial invariant subbundles robustly. The following proposition, which is interesting by itself, plays a key role for the proof of Theorem B. Proposition 1. There exists a C 1 residual subset R 1 ⊂ SRT * such that for every f ∈ R 1 the set of hyperbolic periodic points is dense and it admits a periodic source with dense pre-orbit. Remark 1.1. The statements of Theorem B is a consequence of Theorem 3.2 together with Proposition 1 above. Moreover, one expects that this proposition holds C 1 -generically in RT * provided a counterpart of the connecting lemma (see [5,11]) is given for local diffeomorphisms.
Concerning our results it is an interesting question to understand if there are robustly transitive local diffeomorphisms such that the set of periodic saddle points is dense while the set of periodic sources is non-empty.
The paper is organized as follows. In Section 2 we present some definitions and prove auxiliary lemmas. In Section 3 we prove the main results stated in Section 1. In Section 4 we present a large class of robustly transitive local diffeomorphisms which are not uniformly expanding and exhibit good ergodic properties. Finally, in Section 5 we do some further comments concerning the existence of relevant expanding measures assuming the presence of some kind of dominated splitting.

Robust transitivity and limit sets
In this section we prove some preliminary results relating robust transitivity and existence of dense pre-orbits that play a key role in the proof of the main results. For that purpose we shall introduce first some definitions. Given δ > 0, we say that stands for the ball of radius δ around x. For any endomorphism f : M → M and x ∈ M , the ω-limit set of a point x, denoted by ω f (x), is the set of points y ∈ M such that there exists a sequence (n k ) k∈N of positive integers such that f n k (x) → y when k goes to infinity. Analogously, the α-limit set of x, denoted by α f (x), is the set of accumulation points y ∈ M by the pre-orbit of x, that is, there exists a sequence (x n k ) k∈N in O − f (x) satisfying f n k (x n k ) = x and such that x n k → y when k goes to infinity. Clearly, ω f (x) = M if and only if the forward orbit of x is dense, and analogous statement also holds for pre-orbits. The following lemmas provide a dichotomy of the limit sets for continuous endomorphisms. Proof. Let us suppose that there exists p ∈ M such that O + f (p) is dense in M, otherwise we are done. Write p ℓ = f ℓ (p). Given n ≥ 1 consider the set M n = {x ∈ M : O + f (x) is 1/n-dense}. By assumption, for each ℓ ∈ N and n ∈ N, there is some k n,ℓ such that {p ℓ , · · · , f k n,ℓ (p ℓ )} is 1/2n-dense. Moreover, by continuity of f there exists r n,ℓ > 0 such that f j (B r n,ℓ (p ℓ )) ⊂ B 1/2n (f j (p ℓ )) for all 0 ≤ j ≤ k n,ℓ and, consequently, for any y ∈ B r n,ℓ (p ℓ ) it follows that the finite piece of orbit {y, · · · , f k n,ℓ (y)} is 1/n-dense. Therefore ℓ∈N B r n,ℓ (p ℓ ) ⊂ M n is a open and dense set. In particular this proves that n∈N ℓ∈N B r n,ℓ (p ℓ ) is a residual subset contained Given a continuous endomorphism f ∈ C 0 (M, M ) we denote by C f the set of critical points of f , that is, x ∈ C f if for all r > 0 the restriction f | Br(x) is not a homeomorphism. The next result relates forward and backward limit sets.
Proof. The proof mimics the previous lemma with some care with the critical set C f . Since M \ C f is open and dense, then j≥0 f −j (M \ C f ) is residual. It follows from the previous lemma that the intersection This finishes the proof of the lemma.
In particular we obtain the following immediate consequence: 1. If f ∈ RT then there exists a residual subset of points in M with dense orbit and pre-orbit.
In fact, a converse result also holds obtaining that robust density of points with dense pre-orbit is equivalent to robust transitivity for local diffeomorphisms. Lemma 2.3. Let U be an open subset of the space of C 1 local diffeomorphisms and assume that every f ∈ U admits a dense set of points with dense pre-orbit. Then every f ∈ U is robustly transitive, that is, U ⊂ RT .
Proof. Since the proof is simple we leave it as an easy exercise for the reader.
Let us mention that expanding endomorphisms are not the only class of maps satisfying the assumptions of the previous lemma. In Section 4 we present a class of robustly transitive local diffeomorphisms that are not uniformly expanding but for which there exists a generic subset of points with dense pre-orbit.

Proof of the main results
This section is devoted to the proof of our main results.
3.1. Proof of Theorem A. Items (1) and (2) are a consequence of the C 1 closing lemma for local diffeomorphisms (see e.g. [4,8,10]) and Kupka-Smale theorem for local diffeomorphisms. Hence, there exists a residual subset R 0 ⊂ RT such that for every f ∈ R 0 holds that Per h (f ) = Ω(f ) = M , where Per h (f ) denotes the set of hyperbolic periodic points for f . So, we are left to prove the existence of dense pre-orbits for a generic subset of robustly transitive local diffeomorphisms. Using Corollary 2.1 it follows that every f ∈ R 0 satisfies property (3). This finishes the proof of the theorem.

3.2.
Proof of Proposition 1. Fix f 0 ∈ SRT * . The first step is to recall that f 0 is volume expanding, that is, | det(Df 0 )| > σ > 1. This follows from adapting the arguments used by Bonatti, Díaz and Pujals [1] in the invertible setting, as we can see in the following theorem.
is C 1 robustly transitive set and it has no splitting in a C 1 robust way. Then f is volume expanding.
Since there is no splitting in a C 1 -robust way we can proceed as in [1, Lemma 6.1] to prove that there exists a C 1 local diffeomorphism f ∈ SRT * arbitrarily close to f 0 and a periodic point f k (p) = p such that Df k (p) is an homothety. Moreover, since f satisfies the hypothesis of the theorem above we deduce that f is volume expanding and, consequently, p is periodic repelling and has a dense pre-orbit. Since this is a robust property we deduce that there is an open and dense subset A ⊂ SRT * such that every f ∈ A has a repelling periodic point. In particular, if R 0 is given by Theorem A (adapted RT * with same proof) then every map in the residual subset R 0 ∩ A ⊂ SRT * has a dense set of hyperbolic periodic points and at least one periodic repelling point with dense pre-orbit. This finishes the proof of the proposition.

Proof of Theorem B.
In this section we use the notion of zooming times to deduce the existence of interesting measures. More precisely, we prove the following: Theorem 3.2. If a C 1 local diffeomorphism f has a periodic source with dense preorbit then there are uncountable many invariant, ergodic and expanding measures with full support in Ω(f ) and exponential decay of correlations. In particular, if f is transitive the measure support is total.
In order to prove the previous result we shall adapt some ideas from [9]. In fact, the conclusion above on the existence of many ergodic and expanding probability measures with exponential decay of correlations has been established in Proposition 9.3 and Theorem 5 of [9] for C 1+α −maps admitting critical points. The C 1+α −assumption is used there to obtain bounded distortion under the presence of critical points. Here we prove that this condition can be relaxed for local diffeomorphisms. Let us introduce the zooming times notion and a useful lemma before proving Theorem 3.2. • α n (r) < r, for every r > 0 and n ≥ 1; • α n (r) ≤ α n ( r), for every 0 ≤ r ≤ r and n ≥ 1; • α n • α m (r) ≤ α n+m (r), for every r > 0 and n, m ≥ 1; Observe that an exponential backward contraction is an example of a zooming contraction, α n (r) = λ n r with 0 < λ < 1. Let α = {α n } n be a zooming contraction and δ a positive constant.
Definition 3.2 (Zooming times). We say that n ≥ 1 is a (α, δ)-zooming time for p ∈ M , with respect to f , if there is a neighborhood V n (p) of p such that f n sends Figure 1. A zooming time for x ∈ Z 4 (α, δ, f ) V n (p) homeomorphically onto B δ (f n (p)) and for all x, y ∈ V n (p) and 0 ≤ j < n The ball B δ (f n (p)) is called a zooming ball and the set V n (p) is called a zooming pre-ball. Denote by Z n (α, δ, f ) the set of points of X for which n is an (α, δ)-zooming Moreover, a positively invariant set Λ ⊂ X is called a (α, δ)-zooming set, with respect to f , if every x ∈ Λ is (α, δ)-zooming. Proof. Let γ = period(p). Since O + f (p) is a finite set there exists n 0 ≥ 1 large so that log( (Df n γ (q)) −1 −1 ) > log 32 for all n ≥ n 0 and every q ∈ O + f (p). Let δ > 0 be small enough such that, for every q ∈ O + f (p) there is a neighborhood W (q) of the point q satisfying f n0γ (W (q)) = B δ (q) and (f n0γ | W (q) ) −1 is a (e −λ0 )-contraction, where λ 0 = log 16.
So in the remaining of this section we describe how to construct uncountable many ergodic and expanding measures with full support and exponential decay of correlations.
Proof of Theorem 3.2. Since the proof follows closely the one of [9, Proposition 9.3 ] we give an outline of the proof and focus on the main ingredients. Assume that f is a C 1 local diffeomorphism and p is a periodic source with dense pre-orbit O − f (p). Then, by the previous lemma there exist ℓ ∈ N and δ > 0, such that is also a (α, δ)-zooming set for f ), where α = {α n } n is the zooming sequence given by α n (x) = (1/8) n x. Moreover, changing p for some p ′ ∈ O + f (p) if necessary, we have that O Z (p) := y ∈ O − f (p) ; #{j ∈ N ; y ∈ Z j (α, δ, f ) and f j (y) = p} = ∞ is dense in a neighborhood of p. As n α n (r) < r/4, let 0 < r < δ/4 be small such that B r (p) ⊂ O Z (p).
So, the (α, δ)-zooming nested ball with respect to f , ∆ = B * r (p), is an open neighborhood of p contained in B r (p) (see Definition 5.9 and also Lemma 5.12 in [9] for more details). Furthermore, there is a dense set of points in ∆ (the pre-orbit O Z (p) ∩ ∆) returning by f to ∆ in a (α, δ)-zooming time. In consequence, it follows from [9, Corollary 6.6 ] that there exists collection P of open connected subsets of ∆ and an induced map F : ∆ → ∆ given by F (x) = f R(x) (x) (= f ℓ R(x) (x)), with {R > 0} = P ∈P P , such that R is "the first (α, δ)-zooming return time" to ∆ (see Definition 6.2 and 6.3 of [9]).
The function R : ∆ → N is constant on elements of P and F satisfies the Markov property that F (P ) = ∆, F | P is a C 1 -diffeomorphism and DF | P > 8 for all P ∈ P. Now, for any sequence a = (a P ) P ∈P of real numbers satisfying 0 < a P < 1, is defined on elements of the partition P (n) = n−1 a Pj for all n ≥ 1. It is not hard to check that ν a is a F -invariant and ergodic probability measure and ν a has constant Jacobian on cylinders (in fact J νa F | P = a P for all P ∈ P). Now, using that R dν a = P ∈P a P R(P ) < ∞ then defines an f -invariant and ergodic probability measure. Moreover, µ a | ∆ ≪ ν a and using that ν a gives positive weight to open subsets of ∆ then P ⊂ supp µ a . Since f is transitive every positive invariant set with non empty interior is dense. Thus, µ a has dense support and by compactness, the support is the whole manifold. Furthermore, since each probability measure µ a is ergodic and two ergodic measures either coincide or are mutually singular, we deduce that there are uncountably many ergodic measures with full support, and those measures are expanding. Indeed, ν a -almost every x is (α, δ)-zooming, because ν a j≥0 F −j ({R > 0}) = 1 and we have lim sup for every x ∈ j≥0 F −j ({R > 0}). So, lim n 1 n n−1 j=0 log (D f ( f j (x))) −1 −1 > log 8 > 0 for µ a -almost every x, since µ a is f -invariant (and so, f -invariant). This implies that all Lyapunov exponents with respect to f (and also to f ) are positive for µ a -almost every point. Finally, we notice that by [13], µ a has exponential decay of correlations provided that ν a (R ≥ n) = k≥n R(P )=k a P has exponential decay in n. Since the later property is satisfied for an uncountable many (a P ) P ∈P , this finishes the proof of the theorem.

4.1.
Existence of expanding measures with exponential decay. We shall consider now an important class of robustly transitive local diffeomorphisms introduced in [6,7]. Take n ≥ 2 and r ≥ 1. The following result holds: Main Theorem] Let f ∈ E r (T n ) be volume expanding map such that {w ∈ f −k (x) : k ∈ N} is dense for every x ∈ T n and satisfies the properties: i) There is an open set U 0 in T n such that f| U c 0 is expanding and diam(U 0 )<1; ii) There exists 0 < δ 0 < diam int (U c 0 ) and there exists an open neighborhood U 1 of U 0 such that for every arc γ in U c 0 with diameter larger than δ 0 , there is a point y ∈ γ such that f k (y) ∈ U c 1 for any k ≥ 1; and iii) For every z ∈ U c 1 , there existsz ∈ U c 1 such that f (z) = z.
Then, for every g C r −close enough to f all the points have dense pre-orbit, that is, {w ∈ g −k (x) : k ∈ N} is dense for every x ∈ T n . In particular, f is C r -robustly transitive.
The latter theorem essentially means that robust transitivity is obtained for local diffeomorphisms whose pre-orbits are dense, if it is uniformly expanding in a definite region of the ambient space and for every sufficiently large arc in this expanding region there exists a point whose forward orbit remains in the expanding region. In particular all periodic points have in fact dense pre-orbits. In fact, it follows from [7] that hypotheses (2) and (3) above assure the existence of a locally maximal expanding invariant set Λ f which has a topological property of 'separation'. Roughly, every open set intersects Λ f after a finite number of iterates which implies for future iterates the internal radius growth(IRG property): Note that Lemma 4.1 proves the robustness of IRG property, which is fundamental to prove the density of the pre-orbit of any point under the perturbed map. For further details, see [7]. After the discussion above, we are now in condition to present a large class of examples that illustrate our main results. Let us consider F the class of C r endomorphisms f in the n-dimensional torus T n satisfying the following properties: (1) (volume expanding) There exists σ > 1 such that |det(Df (x))| ≥ σ for all x ∈ T n ; (2) Every point has dense pre-orbit; (3) There is an open set U 0 in T n such that f| U c 0 is expanding and diam(U 0 )<1; (4) There exists a locally maximal expanding invariant set Λ f ⊂ U c 0 with the topological property of 'separation' above. Observe that every map satisfying the assumptions of Theorem 4.1 belong to F . Indeed, property (4) can be shown to be a consequence of the hypotheses (ii) and (iii) of Theorem 4.1 ( we refer the reader to [7] for details), while by property (2) all periodic points have dense pre-orbit. Moreover, since the periodic points are dense, there are plenty of them in the expanding region. Hence, there is at least one periodic source in Λ f , because this set is invariant and expanding. Thus, every f ∈ F has at least one periodic source in Λ f with dense pre-orbit. Therefore, Theorem 3.2 yields for any f ∈ F there are uncountable many ergodic, invariant and expanding measures with full support and exponential decay of correlations.

4.2.
Example of robustly non-existence of splitting. In this subsection we provide a large class of examples satisfying our main results with robust nonexistence of splitting from [7, Example 1]. We provide here just the main ideas about the construction and show that this class of maps satisfy the assumptions of our main results, and refer the reader to [7] for details on the construction.
Let us consider a linear expanding endomorphism E : T n → T n with n ≥ 2 and large topological degree N . There exists a Markov partition, denote the elements by R i with 1 ≤ i ≤ N . For our purpose we can work with the initial partition, but if you wish to construct a more general example doing a deformation of the Markov partition it is enough to consider an isotopic map to the identity.
Pick U 0 an open set in T n such that its convex hull U on the lift is contained in the interior of [0, 1] n (that implies diam(U 0 ) < 1) and there exists at least one R i contained in U c 0 . Assume there exist p ∈ U 0 and q i ∈ U c 0 expanding fixed points of E with 1 ≤ i ≤ n − 1. These requirements are feasible since E has large degree. Choose ε > 0 small enough such that B ε (q i ) ∩ U 0 = ∅ and B ε (q i ) ∩ B ε (q j ) = ∅ for all i = j. Denote the tangent space splitting as follows where ≺ denote that E u i (x) dominates the expanding behavior of E u i−1 (x). We proceed now to deform E by a smooth isotopy supported in U 0 ∪ ( B ε (q i )). The perturbation f is done in such a way that: (1) the continuation of p goes through a pitchfork bifurcation, giving birth to two periodic points r 1 and r 2 in U 0 such that both are repeller, p becomes a saddle point and f still expands volume in U 0 ; (2) two expanding eigenvalues of q i associated to E u i (q i ) and E u i+1 (q i ) of T qi (T n ) become complex expanding eigenvalues for f . Thus, these two expanding subbundles are mixed obtaining T f qi (T n ) = E u 1 ≺ E u 2 ≺ · · · ≺ F u i ≺ · · · ≺ E u n , where F u i is two dimensional and correspond to the complex eigenvalues associated to q i ; Note that the existence of these periodic points with complex eigenvalues prevent any non-trivial invariant subbundle, and this construction is robust. Let us stress that the expanding region in these examples can be taken as small as desired. It can be shown that this class of examples satisfies the assumptions of Theorem 4.1 and, therefore f is a C 1 −local diffeomorphism having a periodic source with dense pre-orbit. In particular, this class of examples admits uncountably many ergodic probability measures with exponential decay of correlations.

Further comments
In this section, we address the problem of existence of relevant expanding measures for robustly transitive local diffeomorphisms that admit some non-trivial invariant subbundle. First we introduce some notions. Given a local diffeomorphism f ∈ Diff loc (M ) and a compact forward invariant set Λ ⊂ M we say that Λ admits a dominated splitting if there exists a continuous splitting T Λ M = E 1 ⊕ E 2 and constants C, a > 0 and λ ∈ (0, 1) such that for all x ∈ Λ and n ∈ N: , and for all v ∈ E 1 x \ {0} and w ∈ C 2 x \ {0} Df n (x)v Df n (x)w ≤ Cλ n .
Since our previous results hold for maps whose tangent bundle does not admit invariant subbundles we now discuss the existence of expanding measures with full support in the presence of dominated splittings that are robust by C 1 -perturbations. We say that a dominated splitting T Λ M = E 1 ⊕ E 2 is of expanding type, namely if the subbundle E 1 satisfies (Df n | E 1 x ) −1 ≤ Cλ n for all x ∈ Λ and n ≥ 1. This implies that f is uniformly expanding and so, by the theory developed by Sinai-Ruelle-Bowen, there are uncountable many f -invariant, ergodic and expanding measures with full support and exponential decay of correlations.
In a dual way, we say that a dominated splitting T Λ M = E 1 ⊕E 2 is of contracting type if the subbundle E 1 satisfies Df n | E 1 x ≤ Cλ n for all x ∈ Λ and n ≥ 1. Note that if T Λ M = E 1 ⊕ E 2 is a dominated splitting of contracting type then expanding measures cannot exist due to the existence of invariant stable direction with uniform contraction along the orbits. Hence, one could ask wether there are uncountable ergodic and hyperbolic measures with total support and exponential decay of correlations. The same strategy to prove Theorem B could answer the previous question provided the existence of Markovian induced schemes for maps with a dense non-uniformly hyperbolic set, which is an open question.
Finally, it remains to consider the case where E 1 is a center bundle with nonuniform expanding or contracting behavior and dominated by a a subbundle E 2 with uniform expansion. Examples illustrating this situation and where there exists a periodic source with dense pre-orbit can be found in [7, subsection 5.3]. In particular such class of maps admit uncountable many invariant, ergodic and expanding probability measures with full support and exponential decay of correlations. We expect an analogous result as Theorem B to hold for these type of maps.
Acknowledgments: The work was initiated after the Workshop on Dynamical Systems-Bahia 2011 at Universidade Federal da Bahia. The authors are grateful to E. Pujals and L. Díaz for useful and encouraging conversations. The first author is grateful to DMAT(PUC-Rio) and ICTP(math section) for the nice environment provided during the preparation of this paper. The first author was supported by CNPq and CDCHT-ULA project number AAA. The second and third authors were also partially supported by CNPq and FAPESB. The second author was partially supported by the Balzan Research Project of J.Palis.