Federico Rodriguez Hertz

• Pennsylvania State University

For a compact negatively curved space, we develop a notion of thermodynamic formalism and apply it to study the space of quasi-morphisms of its fundamental group modulo boundedness. We prove that this space is Banach isomorphic to the space of Bowen functions corresponding to the associated Gromov geodesic flow, modulo a weak notion of Livsic cohom...

We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or $1$-parameter flow and establishes extra...

In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniq...

Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs $\Phi$ on $[0,1]$ where linear and non-linear behaviour coexist. Namely, for every $2\leq r \leq \infty$ we exhibit the existence of a $C^r$-smooth IFS such that $f'\equiv c(\Phi)$ on the attractor and $f''\equiv 0$ for every $f \in \Phi$, yet $\Phi$ is not $C^t$-smooth...

On the analytic side, we prove the quantum ergodicity (QE) of Hamiltonian operators on certain series of unitary flat bundles, using mixed quantization techniques. On the dynamical side, we introduce a new family of partially hyperbolic flow associated with QE and establish its ergodicity.

We prove that every genuinely partially hyperbolic $\mathbb {Z}^r$ -action by toral automorphisms can be perturbed in $C^1$ -topology, so that the resulting action is continuously conjugate, but not $C^1$ -conjugate, to the original one.

Let $\Phi$ be a $C^\omega (\mathbb{C})$ self-conformal IFS on the plane, satisfying some mild non-linearity and irreducibility conditions. We prove a uniform spectral gap estimate for the transfer operator corresponding to the derivative cocycle and every given self-conformal measure. Building on this result, we establish polynomial Fourier decay f...

We study actions by lattices in higher-rank (semi)simple Lie groups on compact manifolds. By classifying certain measures invariant under a related higher-rank abelian action (the diagonal action on the suspension space) we deduce a number of new rigidity results related to standard projective actions (i.e. boundary actions) by such groups. Specifi...

We introduce a notion of a point-wise entropy of measures (i.e., local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simpl...

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with Hölder Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon me...

UDC 515.12 We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7 , No. 1, 49–72 (1987)]. Namely, we show that if two Anosov flow of this kind are C 0 conjugate, then t...

We develop a geometric method to establish the existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular, the entropy maximizing measure and the SRB measure. A characterization of equilibrium states in terms of their disintegrations along stab...

We show that every self conformal measure with respect to a $C^2 (\mathbb{R})$ IFS $\Phi$ has polynomial Fourier decay under some mild and natural non-linearity conditions. In particular, every such measure has polynomial decay if $\Phi$ is $C^\omega (\mathbb{R})$ and contains a non-affine map. A key ingredient in our argument is a cocycle version...

This file is composed of questions that emerged or were of interest during the workshop "Interactions between Descriptive Set Theory and Smooth Dynamics" that took place in Banff, Canada on 2022.

In this paper we prove that for topologically mixing Anosov flows their equilibrium states corresponding to H\"older potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely...

We introduce a notion of a point-wise entropy of measures (i.e local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy and with the Ledrappier-Young local entropy on unstable leaves. We show that the neutralized local entropy must coincide with the two other notions of local entropies, and so all three quan...

Let $X_1^t$ and $X_2^t$ be volume preserving Anosov flows on a 3-dimensional manifold $M$. We prove that if $X_1^t$ and $X_2^t$ are $C^0$ conjugate then the conjugacy is, in fact, smooth, unless $M$ is a mapping torus of an Anosov automorphism of $\mathbb T^2$ and both flows are constant roof suspension flows. We deduce several applications. Among...

We show that the space of expanding maps contains an open and dense subset where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.

We apply the matching functions technique in the setting of contact Anosov flows which satisfy a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman-Ornstein~\cite{FO}. Namely, we show that if two such Anosov flows are $C^0$ conjugate then they are $C^{r}$, conjugate for some $r\in[1,2)$ or even $C^\infty$...

We introduce the matching functions technique in the setting of Anosov flows. Then we observe that simple periodic cycle functionals (also known as temporal distance functions) provide a source of matching functions for conjugate Anosov flows. For conservative codimension one Anosov flows $\varphi^t\colon M\to M$, $\dim M\ge 4$, these simple period...

Let Φ be a C1+γ smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {pkx}k∈N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29...

We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.

In this note we report some advances in the study of ther- modynamic formalism for a class of partially hyperbolic systems—cen- ter isometries—that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying on sym- bolic dynamics) and even provide new information in the classical case. For such s...

In this paper we use the blow-up surgery introduced in [1] to produce new higher dimensional partially hyperbolic flows. The main contribution of the paper is the slow-down construction which accompanies the blow-up construction. This new ingredient allows to dispose of a rather strong domination assumption which was crucial for results in [1]. Con...

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $\mu$. We show that if $f:(M,\mu)\to (M,\mu)$ is exponentially mixing then it is Bernoulli.

In this paper we introduce a new methodology for smooth rigidity of Anosov diffeomorphisms based on "matching functions." The main observation is that under certain bunching assumptions on the diffeomorphism the periodic cycle functionals can provide such matching functions. For example we consider a sufficiently small C^1 neighborhood of a linear...

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying in symbolic dynamics) and even provide new information in the classical case. For such syst...

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with H\"older Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon...

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with H\"older Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon...

We develop a geometric method to establish existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular the entropy maximizing measure and the SRB measure. It is also given a characterization of equilibrium states in terms of their disintegration...

Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the sec...

We introduce a notion of abelian cohomology in the context of smooth flows. This is an equivalence relation which is weaker than the standard cohomology equivalence relation for flows. We develop Livshits theory for abelian cohomology over transitive Anosov flows. In particular, we prove an abelian Livshits theorem for homologically full Anosov flo...

In this paper we use the blow-up surgery introduced in [G] to produce new higher dimensional partially hyperbolic flows. The main contribution of the paper is the slow-down construction which accompanies the blow-up construction. This new ingredient allows to dispose of a rather strong domination assumption which was crucial for results in [G]. Con...

We show that the space of expanding maps contains an open and dense set where smooth conjugacy classes of expanding maps are characterized by the values of the Jacobians of return maps at periodic points.

We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.

It is proved a classification of three dimensional partially hyperbolic diffeomorphisms assuming some rigid hypotheses on the tangent bundle dynamics.

Partial hyperbolicity appeared in the 1960s as a natural generalization of hyperbolicity. In the last 20 years, there has been great activity in this area. Here we survey the state of the art in some related topics, focusing especially on partial hyperbolicity in dimension three. The reason for this is not only that it is the smallest dimension in...

This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributi...

This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributi...

We make a modest progress in the nonuniform measure rigidity program started in 2007 and its applications to the Zimmer program. The principal innovation is in establishing rigidity of large measures for actions of ℤk, k ≥ 2 with pairs of negatively proportional Lyapunov exponents which translates to applicability of our results to actions of latti...

We study skew products where the base is a hyperbolic automorphism of $\mathbb{T}^2$, the fiber is a smooth area preserving flow on $\mathbb{T}^2$ with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation a...

In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{H\"older}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action...

In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{H\"older}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action...

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the mani...

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the mani...

In this paper we give the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle. The existence of such an example had been an open question since 1975 [4]. http://authors.elsevier.com/a/1TKyD12uNpAymd

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the pro...

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $\Gamma$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $\alpha$ be a smooth $\Gamma$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $\rho$ of $\alp...

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $\Gamma$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $\alpha$ be a smooth $\Gamma$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $\rho$ of $\alp...

For a partiallyhyperbolic diffeomorphism on a 3-manifold, we show that any invariant foliation tangent to the center-unstable (or center-stable) bundle has no compact leaves.

In this paper we give the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle. The existence of such an example had been an open question since 1975.

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms.

Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$, we study $\nu$-stationary probability measures on $M$. We prove for hyperbolic stationary measures the following trichotomy: either the stable distributions are non-random, the measure is SRB, or the measure is supported on a finite set and is he...

We consider two numerical entropy--type invariants for actions of $\Zk$, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. $\Zk,\,k\ge 2$ actions on $k+1$-dimensional manifolds. In this case we show that for a fixed dimension (or,...

We prove that any smooth action of $\mathbb Z^{m-1}, m\ge 3$ on an $m$-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e. isomorphic up to a finite permutation to an affine action on the torus or its factor by $\pm\Id$. Furthermore this isomorphism...

We show that all $C^\infty$ Anosov $Z^r$-actions on tori and nilmanifolds without rank-one factor actions are, up to $C^\infty$ conjugacy, actions by automorphisms.

We show that all $C^\infty$ Anosov $Z^r$-actions on tori and nilmanifolds without rank-one factor actions are, up to $C^\infty$ conjugacy, actions by automorphisms.

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the pro...

We show that various classes of closed manifolds with non-trivial higher homotopy groups do not support (transitive) Anosov diffeomorphisms. In particular we show that a finite product of spheres at least one of which is even-dimensional does not support transitive Anosov diffeomorphisms.

In this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected Riemannian manifolds. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets u...

We give a description of ergodic components of SRB measures in terms of ergodic homoclinic classes associated to hyperbolic periodic points. For transitive surface diffeomorphisms, we prove that there exists at most one SRB measure.

In [18] the authors proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stably ergodic dif-feomorphisms are dense among the partially hyperbolic ones and, in subsequent results [20, 21], they obtained a more accurate description of this abundance of ergodicity in dimension three. This work is a s...

Let M be a closed orientable irreducible 3-manifold, and let f be a diffeomorphism over M. We call an embedded 2-torus T an Anosov torus if it is invariant and the induced action of f over \pi_1(T) is hyperbolic. We prove that only few irreducible 3-manifolds admit Anosov tori: (1) the 3-torus, (2) the mapping torus of -id, and (3) the mapping toru...

We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of 3-dimensional manifolds having compact center leaves: either there is a unique entropy maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0 or, there is a finite number of entropy maximizing measures, all of them wit...

We prove that any real-analytic action of SL(n, Z), n ≥ 3 with standard homotopy data that preserves an ergodic measure µ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic...

In this survey we shall present some relations between measure theory and geometric topology in dynamics. One of these relations comes as follows, on one hand from topological information of the system, some structure should be preserved by the dynamics at least in some weak sense, on the other hand, measure theory is soft enough that an invariant...

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance...

In this work we prove that each C^r conservative diffeomorphism with a pair of hyperbolic periodic points of co-index one can be C^1-approximated by C^r conservative diffeomorphisms having a blender.

We prove that stable ergodicity is C r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r∈[2,∞]. The proof follows the Pugh–Shub program [29]: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C r open and dense, and essen...

A. In this paper we will prove some results about integrability of the weak invariant bundles for partially hyperbolic diffeomorphisms in dimension 3. We deal with the problems of existence and uniqueness in case we have transitivity and denseness of periodic orbits. We prove that if we have two crossing central curves contained in a weak-un...

It is shown that stable accessibility property is C r -dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle, for r ≥ 2, volume preserving or not. This establishes a conjecture by Pugh and Shub for these systems.

We consider an ergodic invariant measure μ for a smooth action α of ℤ k ,k≥2, on a (k+1)-dimensional manifold or for a locally free smooth action of ℝ k ,k≥2, on a (2k+1)-dimensional manifold. We prove that if μ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in ℤ k has positive entropy, then μ is absolutely conti...

We consider an ergodic invariant measure μ for a smooth action α of Zk, k ≥ 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of Rk, k ≥ 2, on a (2k + 1)-dimensional manifold. If μ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Zk has positive entropy, then μ is absolutely continuous. Th...

In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer,...

Using the definition of dominated splitting, we introduce the notion of critical set for any dissipative surface diffeomorphism as an intrinsically well-defined object. We obtain a series of results related to this concept.

Every C 2 action α of Z k , k ≥ 2, on the (k + 1)-dimensional torus whose elements are homotopic to the corresponding elements of an action α 0 by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between α and α 0 . This measure is absolutely continuous and the semiconjugacy provides...

Using the definition of dominated splitting, we introduce the no- tion of critical set for any dissipative surface diffeomorphism as an intrinsically well-defined object. We obtain a series of results related to this concept. Using the definition of dominated splitting, we introduce the no- tion of critical set for any dissipative surface diffeomor...

In [15] the authors proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stable ergodic diffeomorphism are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples o...

We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\T^N$.

Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7) Transitivity and (8) Classification. Here we will survey the state of the art on these subjects, and propose rel...

We prove, for f a partially hyperbolic diffeomorphism with center dimension one, two results about the integrability of its central bundle. On one side, we show that if the non wandering set of f is the whole manifold, and the manifold is 3 dimensional, then the absence of periodic points implies the unique integrability of the central bundle. On t...

We prove that a cohomology free flow on a manifold M fibers over a diophantine translation on the torus T b, where b is the first Betti number of M.

We obtain a local topological and dynamical description of expansive attractors on surfaces. The main result is that expansive attractors on surfaces are hyperbolic and have local product structure, except possibly at a finite number of periodic points, which can be either sinks, singularities or épines. Some open questions concerning this kind of...

We find a class of ergodic linear automorphisms of T-N that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of T-N are stably ergodic when N <= 5.

We find a class of ergodic linear automorphisms of TN that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of TN are stably ergodic when N ¡Ü 5.

Despite its homotopical stability, new relevant dynamics appear in the isotopy class of a pseudo-Anosov homeomorphism. We study these new dynamics by identifying homotopically equivalent orbits, obtaining a more complete description of the topology of the corresponding quotient spaces, and their stable and unstable sets. In particular, we get some...

In this paper we prove that any C^1 vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C^1 -flows on three manifolds. For that...

Let $S$ be a surface of nonpositive curvature of genus bigger than 1 (i.e. not the torus). We prove that any flat strip in the surface is in fact a flat cylinder. Moreover we prove that the number of homotopy classes of such flat cylinders is bounded.

We prove that some ergodic linear automorphisms of $\T^N$ are stably ergodic, i.e. any small perturbation remains ergodic. The class of linear automorphisms we deal with includes all non-Anosov ergodic automorphisms when N=4 and so, as a corollary, we get that every ergodic linear automorphism of $\T^N$ is stably ergodic when $N\leq 5$.

We prove that some ergodic linear automorphisms of T are stably ergodic, i.e. any small perturbation remains ergodic. The class of linear automorphisms we deal with includes all non-Anosov ergodic automorphisms when N = 4 and so, as a corollary, we get that every ergodic linear automorphism of T N is stably ergodic when N 5. 1.

In this paper we study entropy maximizing measures for 3-dimensional accessible partially hyperbolic diffeomorphisms with compact center one-dimensional leaves. We obtain the following dichotomy: either there is a unique entropy maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0 or, there are at least...

In this work we obtain new criteria for ergodicity of di eomorphisms involving conditions on Lyapunov exponents, existence of blenders, dominated splittings and periodic points. These criteria allow us to prove (Theorem A) the abundance in the C1 topology of stably ergodic di eomorphisms among the conservative partially hyperbolic ones provided the...