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Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Nalini Anantharaman, Stéphane Nonnenmacher (2007)

Annales de l’institut Fourier

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

Hausdorff and packing dimensions for ergodic invariant measures of two-dimensional Lorenz transformations

Franz Hofbauer (2009)

Commentationes Mathematicae Universitatis Carolinae

We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence...

Heteroclinic solutions for perturbed second order systems

Massimiliano Berti (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The existence of infinitely many heteroclinic orbits implying a chaotic dynamics is proved for a class of perturbed second order Lagrangian systems possessing at least 2 hyperbolic equilibria.

Heterodimensional cycles, partial hyperbolicity and limit dynamics

L. J. Diaz, J. Rocha (2002)

Fundamenta Mathematicae

We study one-parameter families of diffeomorphisms unfolding heterodimensional cycles (i.e. cycles containing periodic points of different indices). We construct an open set of such arcs such that, for a subset of the parameter space with positive relative density at the bifurcation value, the resulting nonwandering set is the disjoint union of two hyperbolic basic sets of different indices and a strong partially hyperbolic set which is robustly transitive. The dynamics of the diffeomorphisms we...

High-order phase transitions in the quadratic family

Daniel Coronel, Juan Rivera-Letelier (2015)

Journal of the European Mathematical Society

We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as x ↦ exp ( – x – 2 ) near x = 0 , before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

Homology of origamis with symmetries

Carlos Matheus, Jean-Christophe Yoccoz, David Zmiaikou (2014)

Annales de l’institut Fourier

Given an origami (square-tiled surface) M with automorphism group Γ , we compute the decomposition of the first homology group of M into isotypic Γ -submodules. Through the action of the affine group of M on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.

Hopf bifurcation analysis of some hyperchaotic systems with time-delay controllers

Lan Zhang, Cheng Jian Zhang (2008)

Kybernetika

A four-dimensional hyperchaotic Lü system with multiple time-delay controllers is considered in this paper. Based on the theory of Hopf bifurcation in delay system, we obtain a simple relationship between the parameters when the system has a periodic solution. Numerical simulations show that the assumption is a rational condition, choosing parameter in the determined region can control hyperchaotic Lü system well, the chaotic state is transformed to the periodic orbit. Finally, we consider the differences...

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