Related papers: A $C^{r}$ Closing Lemma for a Class of Symplectic …

We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all…

We show that any area-preserving $C^r$-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be $C^r$-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every $1\le r\le \infty$.…

We study the problem of conjugating a diffeomorphism of the interval to (positive) powers of itself. Although this is always possible for homeomorphisms, the smooth setting is rather interesting. Besides the obvious obstruction given by…

In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual…

Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\mathbb{D}^{2}$, allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the…

We show that, for every compact n-dimensional manifold, n\geq 1, there is a residual subset of Diff^1(M) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either…

In this paper we prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.

We show that C^r generically in the space of C^r conservative diffeomorphisms of a compact surface, every hyperbolic periodic point has a transverse homoclinic orbit

We prove the existence of infinitely many periodic points of symplectomorphisms isotopic to the identity if they admit at least one (non-contractible) hyperbolic periodic orbit and satisfy some condition on its flux. The obtained periodic…

Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $\omega$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$…

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there…

Let Diff^1(M) be the set of all C^1-diffeomorphisms f : M \rightarrow M, where M is a compact boundaryless d-dimensional manifold, d \geq 2. We prove that there is a residual subset R of Diff^1(M) such that if f \in R and if H(p) is the…

For any symplectic manifold, Hamiltonian diffeomorphism group contains a subset which consists of times one flows of autonomous(time-independent) Hamiltonian vector fields. Polterovich and Shelukhin proved that the complement of autonomous…

We provide an affirmative answer to the Cr Closing Lemma, r>1, for a large class of flows defined on every closed surface.

We prove that every $C^2$ conservative partially hyperbolic diffeomorphism of a closed 3-manifold without periodic points is ergodic, which gives an affirmative answer to the Ergodicity Conjecture by Hertz-Hertz-Ures in the absence of…

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational…

We propose a general framework, within which we prove that several properties, such as the fast growth of the number of periodic points, the universality, and the high emergence, hold true for every parameter value for a generic…

We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex…

We prove that every dynamically coherent plaque expansive partially hyperbolic diffeomorphism is topologically stable with respect to the central foliation (in short, {\em plaque topologically stable}). Next, we study partially hyperbolic…

In this work one proves that, around each point of a dense open set (regular points), a real analytic or holomorphic bihamiltonian structure decomposes into a product of a Kronecker bihamiltonian structure and a symplectic one if a…