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Jorge Rocha

Publications and source records attributed to Jorge Rocha.

10 recordsLinked to original sources
arXiv2017

The centralizer of Komuro-expansive flows and expansive R^d-actions

In this paper we study the centralizer of flows and $\mathbb R^d$-actions on compact Riemannian manifolds. We prove that the centralizer of every $C^\infty$ Komuro-expansive flow with non-ressonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that $\mathbb R^d$-actions obtained as suspension of $\mathbb Z^d$-actions are expansive if and only if the same holds for the $\mathbb Z^d$-actions. We also show that homogeneous expansive $\mathbb R^d$-actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparametrizations of the $\mathbb R^d$-action. In particular, homogeneous Anosov $\mathbb R^d$-actions have quasi-trivial centralizer.

math.DS↗
arXiv2017

On sensitivity to initial conditions and uniqueness of conjugacies for structurally stable diffeomorphisms

In this paper we study $C^1$-structurally stable diffeomorphisms, that is, $C^1$ Axiom A diffeomorphisms with the strong transversality condition. In contrast to the case of dynamics restricted to a hyperbolic basic piece, structurally stable diffeomorphisms are in general not expansive and the conjugacies between $C^1$-close structurally stable diffeomorphisms may be non-unique, even if there are assumed $C^0$-close to the identity. Here we give a necessary and sufficient condition for a structurally stable diffeomorphism to admit a dense subset of points with expansiveness and sensitivity to initial conditions. Morever, we prove that the set of conjugacies between elements in the same conjugacy class is homeomorphic to the $C^0$-centralizer of the dynamics. Finally, we use this fact to deduce that any two $C^1$-close structurally stable diffeomorphismsare conjugated by a unique conjugacy $C^0$-close to the identity if and only if these are Anosov.

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arXiv2016

The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial

In the late nineties, Smale proposed a list of problems for the next century and, among these, it was conjectured that for every $r\ge 1$ a $C^r$-generic diffeomorphism has trivial centralizer. Our contribution here is to prove the triviality of $C^r$-centralizers on hyperbolic basic sets. In particular, $C^r$-generic transitive Anosov diffeomorphisms have a trivial $C^1$-centralizer. These results follow from a more general criterium for expansive homeomorphisms with the gluing orbit property. We also construct a linear Anosov diffeomorphism on $\mathbb T^3$ with discrete, non-trivial centralizer and with elements that are not roots. Finally, we prove that all elements in the centralizer of an Anosov diffeomorphism preserve some of its maximal entropy measures, and use this to characterize the centralizer of linear Anosov diffeomorphisms on tori.

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arXiv2016

Uniform hyperbolicity revisited: Index of periodic points and equidimensional cycles

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 diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the C1-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak*-topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The later can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.

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arXiv2015

Generic Hamiltonian Dynamics

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a C2-residual R in the set of C2 Hamiltonians on a closed symplectic manifold M, such that, for any H in R, there is a full measure subset of energies e in H(M) such that the Hamiltonian level (H,e) is topologically mixing; moreover these level sets are homoclinic classes.

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arXiv2010

Topological stability for conservative systems

We prove that the C1-interior of the set of all topologically stable C1-incompressible flows is contained in the set of Anosov incompressible flows. Moreover, we obtain an analogous result for the discrete-time case.

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arXiv2009

On the Stability of the Set of Hyperbolic Closed Orbits of a Hamiltonian

A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e \in \mathbb{R}$, is said to be Anosov if there exists a connected component $\mathcal{E}_{H,e}$ of $H^{-1}({e})$ which is uniformly hyperbolic for the Hamiltonian flow $X_H^t$. The pair $(H,e)$ is said to be a Hamiltonian star system if there exists a connected component $\mathcal{E}^\star_{H,e}$ of the energy level $H^{-1}({e})$ such that all the closed orbits and all the critical points of $\mathcal{E}^\star_{H,e}$ are hyperbolic, and the same holds for a connected component of the energy level $\tilde{H}^{-1}({\tilde{e}})$, close to $\mathcal{E}^\star_{H,e}$, for any Hamiltonian $\tilde{H}$, in some $C^2$-neighbourhood of $H$, and $\tilde{e}$ in some neighbourhood of $e$. In this article we prove that for any four-dimensional Hamiltonian star level $(H,e)$ if the surface $\mathcal{E}^\star_{H,e}$ does not contain critical points, then $X_H^t|_{\mathcal{E}^\star_{H,e}}$ is Anosov; if $\mathcal{E}^\star_{H,e}$ has critical points, then there exists $\tilde{e}$, arbitrarily close to $e$, such that $X_H^t|_{\mathcal{E}^\star_{H,\tilde{e}}}$ is Anosov.

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arXiv2008

Contributions to the Geometric and Ergodic Theory of Conservative Flows

We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

math.DS↗
arXiv2006

Removing zero Lyapunov exponents in volume-preserving flows

Baraviera and Bonatti proved that it is possible to perturb, in the c^1 topology, a volume-preserving and partial hyperbolic diffeomorphism in order to obtain a non-zero sum of all the Lyapunov exponents in the central direction. In this article we obtain the analogous result for volume-preserving flows.

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