Isabel Rios

The boundary of hyperbolicity for Hénon-like families

We consider two-dimensional Henon-like families of difieomorphisms Ha;b depending on parameters a;b. We study the hyperbolicity of the horse- shoe which exists for large values of the parameter a and show that this hy- perbolicity is preserved uniformly until a flrst tangency takes place.

Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes

In this paper, we propose a model for the destruction of three-dimensional horseshoes via heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call generating .

Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies

We study the destruction of hyperbolic sets (horseshoes) in parametrized families of diffeomorphisms through homoclinic tangencies taking place inside the limit set. If the limit set at the tangency parameter has small dimension (limit capacity) then hyperbolicity prevails after the bifurcation (full Lebesgue density). We also prove that, if that limit set is thick then the system exhibits homoclinic tangencies for a whole parameter interval across the bifurcation. These results are based on a geometric analysis of the limit set at the tangency, including a statement of bounded distortion.

Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes

In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the invariant manifolds theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated with Holder continuous potentials.

On t-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes

In this paper we consider horseshoes with homoclinic tangencies inside the limit set. For a class of such maps, we prove the existence of a unique equilibrium state μt, associated to the (non-continuous) potential −t log Ju. We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t0 such that the pressure of μt0 is zero. To deal with the discontinuity of the jacobian, we introduce a countable Markov partition adapted to the dynamics, and work with the first return map defined in a rectangle of it. AMSC : 37C29, 37C45, 37D25, 37D35.

Uniform hyperbolicity for random maps with positive Lyapunov exponents

We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.

Attractors and Repellers Near Generic Elliptic Points of Reversible Maps

We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.

Critical saddle-node horseshoes: bifurcations and entropy

In this paper we study one-parameter families (f?)?[?1,1] of two-dimensional diffeomorphisms unfolding critical saddle-node horseshoes (say at ? = 0) such that f? is hyperbolic for negative ?. We describe the dynamics at some isolated secondary bifurcations that appear in the sequel of the unfolding of the initial saddle-node bifurcation. We construct two classes of open sets of such arcs. For the first class, we exhibit a collection of parameter intervals In, In?(0,1], converging to the saddle-node parameter, In?0, such that the topological entropy of f? is a constant hn in In and hn is an increasing sequence. So, for parameters in In, the topological entropy is upper bounded by the entropy of the initial saddle-node diffeomorphisms. This illustrates the following intuitive principle: a critical cycle of an attracting saddle-node horseshoe is a destroying dynamics bifurcation. In the second class, the entropy of f? does not depend monotonically on the parameter ?. Finally, when the saddle-node horseshoe is not an attractor, we prove that the entropy may increase after the bifurcation.