Related papers: Generic bi-Lyapunov stable homoclinic classes

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of…

We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable…

For $C^1$ diffeomorphisms with continuous invariant splitting without domination, we prove the existence of (un)stable manifold under the hyperbolicity of invariant measures.

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…

Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic diffeomorphism and let $\Gamma$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_{\Gamma}$ is either hyperbolic attracting or rationally…

We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is H\"{o}lder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to…

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We…

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same…

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…

We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.

Given an embedded stable hypersurface in a four-dimensional symplectic manifold, we prove that it is stable isotopic to a $C^0$-close stable hypersurface with the following property: $C^\infty$-nearby hypersurfaces are generically unstable.…

For a generic conservative diffeomorphism of a 3-manifold M, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is non-uniformly hyperbolic and…

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We…

This paper deals with asymptotic stability of a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical…

We push forward the study of higher dimensional stable Hamiltonian topology by establishing two non-density results. First, we prove that stable hypersurfaces are not $C^3$-dense in any isotopy class of embedded hypersurfaces on any ambient…

We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about…

We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that if such a homoclinic class contains a volume-expanding periodic point, then, $C^1$-generically, it…

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…

In this work we deal with partially hyperbolic diffeomorphisms whose central direction is two dimensional. We prove that in general the accessibility classes are immersed manifolds. If, furthermore, the diffeomorphism is dynamically…

We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov…