Related papers: $C^r$-Chain closing lemma for certain partially hy…

We show that for any $C^1$ partially hyperbolic diffeomorphism, there is a full volume subset, such that any Cesaro limit of any point in this subset satisfies the Pesin formula for partial entropy. This result has several important…

We prove that, for $C^1$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from 0, then $H(p)$ must be (uniformly) hyperbolic. This is in sprit of the works…

Pugh and Shub have conjectured that essential accessibility implies ergodicity, for a $C^2$, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satsified by all…

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…

For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the…

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving…

We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…

We are interested in finding a dense part of the space of $C^1$-diffeomorphisms which decomposes into open subsets corresponding to different dynamical behaviors: we discuss results and questions in this direction. In particular we present…

We prove that various classical conformal diffeomorphism groups, which are known to be essential [1], are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological…

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…

We show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non-virtually solvable. In particular we prove that a non-virtually solvable…

Let $f: M \to M$ be a $C^r$-diffeomorphism, $r\geq 1$, defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$, and let $H(p)$ be the homoclinic class associated to the hyperbolic periodic point $p$. We prove that if there…

Let $X$ be a compact metric space and $f:X\to X$ a homeomorphism on $X$. We construct a fundamental domain for the set with finite peaks for each cocycle induced by $\phi\in C(X,R)$. In particular we prove that if a partially hyperbolic…

Motivated by the ergodic closing lemma of Ma\~n\'e, we investigate the $C^\infty$ closing lemma in higher-dimensional Hamiltonian systems, with a focus on the statistical behavior of periodic orbits generated by $C^\infty$-small…

In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with…

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a $C^{1+\alpha}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property…

The statistical properties of mostly expanding partially hyperbolic diffeomorphisms have been substantially studied. In this paper, we would like to address the entropy properties of mostly expanding partially hyperbolic diffeomorphisms. We…

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1, 2, ..., \infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting…

We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set K is contained in a locally invariant center submanifold if and only if each strong stable…

A diffeomorphism f is called super exponential divergent if for every r>1, the lower limit of #Per_n(f)/r^n diverges to infinity as n tends to infinity, where Per_n(f) is the set of all periodic points of f with period n. This property is…