Related papers: $\mathcal{C}^2$ surface diffeomorphisms have symbo…
In this paper, we continue our investigation on sub-additive pressures for $C^1$-smooth partially hyperbolic diffeomorphisms. Under the assumption of unstable almost product property, we show that the unstable Bowen topological pressure on…
We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of…
We prove that for a polynomial diffeomorphism of C^2 , the support of any invariant measure, apart from a few obvious cases, is contained in the closure of the set of saddle periodic points.
In the theory of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, it is possible to compute a one-dimensional representative map for any irreducible isotopy class. The topological entropy of this graph representative…
We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^2$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that…
Given a closed orientable surface (\Sigma) of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on (\Sigma) and the convex compact set of additive functions on…
Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as…
We prove the Widom-Sobolev formula for the asymptotic behaviour of truncated Wiener-Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum…
We prove some generic properties for $C^r$, $r=1, 2, ..., \infty$, area-preserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the…
We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps…
A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the…
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 $CC^{1+a}$, $a>0$,…
By using the variational approach, we prove the existence of Sinai-Ruelle-Bowen measures for partially hyperbolic $\mathcal C^1$ diffeomorphisms with mostly expanding properties. The same conclusion holds true if one considers a dominated…
We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures…
This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and…
Let $f:M\to M$ be a $C^1$ map of a compact manifold $M$, with dimension at least $2$, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume…
Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of…
We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine…
Given an isotopy class between two closed hyperbolic surfaces, the Douady--Earle extension provides a unique analytic diffeomorphism representative. In this paper we investigate the Jacobian of the Douady--Earle extension map $F$. We prove…
We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguy\^en and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted…