On mostly expanding diffeomorphisms

Abstract

We study how physical measures vary with the underlying dynamics in the open class of C r , r > 1, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.Résumé. Nousétudions comment les mesures physiques varient avec la dynamique sousjacente, dans la classe ouverte des difféomorphismes C r , r > 1, fortement partiellement hyperboliques pour lesquelles les exposants de Lyapunov centraux de tout u-état de Gibbs sont positifs. Lorsque transitifs, de tels difféomorphismes possédent une unique mesure physique qui persiste et varie continment avec la dynamique.Un des ingrédients principaux de la preuve est un nouveau lemme de type Pliss qui, appliqué dans le contexte adéquate, implique que la fréquence des temps hyperboliques est proche de un. Une autre nouveauté est l'introduction d'une nouvelle caractérisation des cu-états de Gibbs. Chacun de ses deux aspects ayant leur propre intérêt.Le cas non transitif est aussi traité : dans ce contexte, le nombre de mesures physiques est une fonction semi-continue supérieure du difféomorphisme, et les mesures physiques varient continument sous des hypothèses naturelles.

SRB measures for some diffeomorphisms with dominated splittings as zero noise limits

“…Recall that in both works, the authors make effort to show the existence of positive Lebesgue measure set of (weakly) nonuniformly expanding points, then in terms of the previous techniques from [2] or [1] to find SRB measures. In §6, as a byproduct of our result, we provide a proof of the existence of SRB measures for systems considered in [21] (it works for [4] with simpler arguments). We would like to mention that by our new arguments the partially hyperbolic splittings can be restricted to attractors, rather than the whole manifold, which is pivotal there to find the (weakly) non-uniformly expanding points.…”

“…It turns out that Gibbs u-states and Gibbs cu-states are crucial candidates of SRB measures and physical measures(e.g. [1,10,12,4,21] In this paper, we deal with the problem of the existence of SRB measures for some systems exhibitting dominated splitting. The key idea is to add small random noise to the deterministic dynamical system and prove that as noise levels tends to zero, the limit of the ergodic stationary measures, called the randomly ergodic limit (see Definition 3.4), has ergodic components to be Gibbs cu-states associated to some sub-bundle E whenever this randomly ergodic limit appears some weak expansion along E. Under some extra assumptions on the other directions of the sub-bundles, one can make these Gibbs cu-states to be SRB measures or physical measures.…”

On the non-robustness of intermingled basins

“…First of all, observe that neither µ nor ν can have positive center Lyapunov exponent. This is a consequence of the well-known fact that under our hypotheses the basin of attraction of such a measure would be essentially open (See for instance [7] where the conservative case is discussed with details and recently [1] for a discussion about the non conservative case.) .…”

“…[KD2] For p, q ∈ T 2 , fixed points of A, we assume that the map ψ(p, ·) : [0, 1] → [0, 1] has exactly two fixed points, a source at t = 1 and a sink in t = 0. Analogously, the map ψ(q, ·) : [0, 1] → [0, 1] has exactly two fixed points, a sink at t = 1 and a source in t = 0.…”

Partially volume expanding diffeomorphisms

“…Several other results on the physical measures of partially hyperbolic maps have been obtained, especially in the setting of maps mostly expanding cente that was introduced by Alves, Bonatii, Viana [1]. Andersson, Vásquez [3] use a slightly stronger definition, which they prove is C 2 open. The latter was improved by Yang [27], who proved C 1 openness.…”