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New criteria for ergodicity and nonuniform hyperbolicity. (English) Zbl 1290.37011
In the mid-nineties of the past century, Pugh and Shub conjectured that stable ergodicity is dense among partial hyperbolic dynamical systems. This paper is a remarkable contribution to this conjecture for the discrete-time case.
We recall that for \(\alpha>0\), a \(C^{1+\alpha}\) volume-preserving diffeomorphism is said to be stably ergodic if the exists a \(C^1\)-neighborhood on which every \(C^{1+\alpha}\) volume-preserving diffeomorphism is ergodic. Apart of the long-time known Anosov diffeomorphisms, the first examples of stably ergodic diffeomorphisms outside the hyperbolic world were given in [M. Grayson et al., Ann. Math. (2) 140, No. 2, 295–329 (1994; Zbl 0824.58032)] and were defined as the time-one of a geodesic flow on a negative curvature surface. In this example the neutral flow direction generates a non-hyperbolic one-dimensional manifold (central manifold). The contributions towards the previous conjecture always assumed a “one-dimensionality” hypothesis on the central bundle. Indeed, we have the result [F. Rodriguez Hertz et al., Invent. Math. 172, No. 2, 353–381 (2008; Zbl 1136.37020)] and also [K. Burns and A. Wilkinson, Ann. Math. (2) 171, No. 1, 451–489 (2010; Zbl 1196.37057)] where it is considered that the central bundle displays the “bunching condition” (in rough terms, the central dynamics is conformal, in particular, trivially satisfied if the central bundle is one-dimensional).
In the present paper it is proved that (Theorem B), within the volume-preserving discrete-time setting, stable ergodicity is \(C^1\)-dense among partial hyperbolic dynamical systems with a two-dimensional central bundle. In order to obtain this result, the authors establish a new criterion for ergodicity and for the absence of zero Lyapunov exponents (Theorem A). This criterion assures, in brief terms, ergodicity and non-uniform hyperbolicity under the hypothesis of positive measure of stable/unstable-saturated subsets associated to ergodic homoclinic classes. It is interesting to observe that another concept used along the arguments to obtain stable ergodicity is the concept of “blenders” that was originally introduced to obtain robust transitivity.

MSC:
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D30 Partially hyperbolic systems and dominated splittings
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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