Related papers: Symbolic dynamics for surface diffeomorphisms with…

We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant…

In this paper, we study Borel probability measures of maximal entropy for analytic subsets in a dynamical system. It is well known that higher smoothness of the map over smooth space plays important role in the study of invariant measures…

We prove three formulas for computing topological pressure of $C^1$-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that…

Let $f$ be a non-invertible $C^{1+\beta}(\beta>0)$ map with zero Lyapunov exponents and singularities on a closed Riemannian manifold $M$. We consider the symbolic dynamics of $f$. Combining the techniques in recent works of Sarig, Ovadia…

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic…

We show that every codimension one partially hyperbolic diffeomorphism must support on $\mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is…

Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>\lambda_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$…

We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with…

In this work, we investigate diffeomorphisms whose positiveness of topological entropy is destroyed by singular suspensions. We show that this phenomenon is rare in the set of $C^1$-diffeomorphisms. Precisely, we prove that for an open and…

We prove that for ergodic measures with large entropy have long unstable manifolds for $C^\infty$ surface diffeomorphisms. Specifically, for any $\alpha>0$, there exist constants $\beta>0$ and $c>0$ such that for every ergodic measure $\mu$…

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…

In this note we give examples of Hamiltonian diffeomorphisms which are on one hand dynamically complicated, for instance with positive topological entropy, and on the other hand minimal from the perspective of Floer theory. The minimality…

Let $X$ be a compact complex manifold of dimension $k$ and $f:X \longrightarrow X$ be a dominating meromorphic map. We generalize the notion of topological entropy, by defining a quantity $h_{(m,l)}^{top}(f)$ which measures the action of…

If a $C^{1 + a}$, $a >0$, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. We also give a direct proof of ergodicity of volume-preserving $CC^{1+a}$, $a>0$,…

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional,…

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is…

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact Riemannian manifold and $\mu$ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential $\phi$ there exists a sequence of basic sets $\Omega_n$…

Let $f$ be a $C^{1}$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$ where $E^{s}$ is uniformly…

A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a closed surface…

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$…