Newhouse phenomenon and homoclinic classes

We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources

Entropy along expanding foliations

The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism (\C^1 topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of \C^{1+} partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the \C^1 topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are \C^1 open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for $C^1$ residual subset of diffeomorphisms are discussed. We also provide a new class of robustly transitive diffeomorphisms: every $C^2$ volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is $C^1$ robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving).Comment: This is an improved version, here we add two new applications: Corollary E and Theorem

Cherry flow: physical measures and perturbation theory

In this article we consider Cherry flows on torus which have two singularities: a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by R. Saghin and E. Vargas in~\cite{SV}. We also show that the perturbation of Cherry flow depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to the following three cases: (a) has a saddle connection; (b) a Cherry flow; (c) a Morse-Smale flow whose non-wandering set consists two singularities and one periodic sink. In contrary, when the divergence is non-negative, this flow can be approximated by non-hyperbolic flow with arbitrarily larger number of periodic sinks

Geometrical and measure-theoretic structures of maps with mostly expanding center

In this paper we study physical measures for \C^{1+\alpha} partially hyperbolic diffeomorphisms with mostly expanding center. We show that every diffeomorphism with mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology. Moreover, for each diffeomorphism with mostly expanding center, there exists a $C^1$ neighborhood, such that diffeomorphism among a $C^1$ residual subset of this neighborhood admits finitely many physical measures, whose basins have full volume. We also show that the physical measures for diffeomorphisms with mostly expanding center satisfy exponentially decay of correlation for any H\"older observes

Lyapunov stable chain recurrent classes

We show that for a $C^1$ residual subset of diffeomorphisms far away from homoclinic tangency, the stable manifolds of periodic points cover a dense subset of the ambient manifold. This gives a partial proof to a conjecture of C. Bonatti

Invariance principle and rigidity of high entropy measures

A deep analysis of the Lyapunov exponents, for stationary sequence of matrices going back to Furstenberg, for more general linear cocycles by Ledrappier and generalized to the context of non-linear cocycles by Avila and Viana, gives an invariance principle for invariant measures with vanishing central exponents. In this paper, we give a new criterium formulated in terms of entropy for the invariance principle and in particular, obtain a simpler proof for some of the known invariance principle results. As a byproduct, we study ergodic measures of partially hyperbolic diffeomorphisms whose center foliation is 1-dimensional and forms a circle bundle. We show that for any such C2 diffeomorphism which is accessible, weak hyperbolicity of ergodic measures of high entropy implies that the system itself is of rotation type

Generic Continuity of Metric Entropy for Volume-preserving Diffeomorphisms

Let $M$ be a compact manifold and $\text{Diff}^1_m(M)$ be the set of $C^1$ volume-preserving diffeomorphisms of $M$. We prove that there is a residual subset $\mathcal {R}\subset \text{Diff}^1_m(M)$ such that each $f\in \mathcal{R}$ is a continuity point of the map $g\to h_m(g)$ from $\text{Diff}^1_m(M)$ to $\mathbb{R}$, where $h_m(g)$ is the metric entropy of $g$ with respect to volume measure $m$

Measure-theoretical properties of center foliations

Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add to this theory by constructing stable examples for which the disintegration is singular without being atomic. In the context of diffeomorphisms with mostly contracting center direction, for which upper leafwise absolute continuity is known to hold, we provide examples where the center foliation is not lower leafwise absolutely continuous

Physical measures for the geodesic flow tangent to a transversally conformal foliation

We consider a transversally conformal foliation $\mathcal{F}$ of a closed manifold $M$ endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either there is a transverse holonomy-invariant measure for $\mathcal{F}$, or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin cover a set full for the Lebesgue measure. We also give necessary and sufficient conditions for the foliated geodesic flow to be partially hyperbolic in the case where the foliation is transverse to a projective circle bundle over a closed hyperbolic surface.Comment: 29 pages, final version, to appear in Annales de l'Institut Poincar\'e C, Analyse non lin\'eair

Continuity of topological entropy for perturbations of time-one maps of hyperbolic flows

We consider a $C^1$ neighborhood of the time-one map of a hyperbolic flow and prove that the topological entropy varies continuously for diffeomorphisms in this neighborhood. This shows that the topological entropy varies continuously for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center bundle