Related papers: $\mathcal{C}^2$ surface diffeomorphisms have symbo…

For a surface diffeomorphism, a compact invariant locally maximal set $W$ and some subset $A\subset W$ we study the $A$-exceptional set, that is, the set of points whose orbits do not accumulate at $A$. We show that if the Hausdorff…

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the…

An extension theorem for holomorphic mappings between two domains in $\mathbb C^2$ is proved under purely local hypotheses.

Topological entropy is a measure of complex dynamics. In this regard, multimodal maps play an important role when it comes to study low-dimensional chaotic dynamics or explain some features of higher dimensional complex dynamics with…

We consider volume-preserving flows $(\Phi^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(\Phi^f_t)_{t\in\mathbb{R}}$ has the form $\Phi^f_t(x,y)=(\phi_tx,y+\int_0^t…

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical…

On a one-sided shift of finite type we prove that for a generic Holder continuous function there is a unique maximizing measure. We show that b-Holder continuous functions can be approximated in the a-Holder topology, a<b, by a function…

We give sufficient conditions for a diffeomorphism of a compact surface to be robustly $N$-expansive and cw-expansive in the $C^r$-topology. We give examples on the genus two surface showing that they need not to be Anosov diffeomorphisms.…

In this paper we prove that for an ergodic hyperbolic measure $\omega$ of a $C^{1+\alpha}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $\omega$-full measured set $\widetilde{\Lambda}$ such that for every invariant…

Let f be a proper holomorphic mapping between bounded domains D and D' in C^2. Let M, M' be open pieces on the boundaries of D and D' respectively, that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f…

We prove existence of maximal entropy measures for an open set of non-uniformly expanding local diffeomorphisms on a compact Riemannian manifold. In this context the topological entropy coincides with the logarithm of the degree, and these…

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under…

We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic…

In this paper we study the thermodynamic formalism of strongly transitive endomorphisms $f$, focusing on the set all expanding measures. In case $f$ is a non-flat $C^{1+}$ map defined on a Riemannian manifold, these are invariant…

We construct examples of volume-preserving uniquely ergodic (and hence minimal) real-analytic diffeomorphisms on odd-dimemsional spheres

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is…

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study…

We prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for…

We extend the results of arXiv:2206.08295v2 by showing that any homothety in $\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small…

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…