Rafael Labarca

Stability of singular horseshoes

We present an example of a structural stable vector field on the unit disk D3 ⊂ R3, kangent to the boundary of D3, whose nonwandering set is nonhyperbolic. For this we introduce the concept of singular horseshoe which turns out to be one of the models for structural stability on manifolds with boundary.

The explosion of singular cycles

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Bifurcation of the essential dynamics of Lorenz maps and applications to Lorenz-like flows: Contributions to the study of the expanding case

In this article we provide, by using kneadings sequences, the combinatorial bifurcation diagramme associated to a typical two parameter of Lorenz maps on the real line. We apply these results to two parameter families of geometric Lorenz-like flows.

BIFURCATION OF THE ESSENTIAL DYNAMICS OF LORENZ MAPS ON THE REAL LINE AND THE BIFURCATION SCENARIO FOR LORENZ LIKE FLOWS: THE CONTRACTING CASE

In this article we provide, by using kneading sequences, the combinatorial bifurcation diagram associated to a typical two parameter family of contracting Lorenz maps on the real line. We apply these results to two parameter families of geometric Lorenz-like flows.

Stability of Morse-Smale vector fields on manifolds with boundary

LET A4 be a compact manifold with boundary aM and denote by %^‘( M, JM) the space of C’ vector fields on M, that are tangent to dM endowed with the usual C’ topology. In this space it is natural to define structural stability as in the boundaryless case, namely saying that X E a’( M, c?M) is C’ structurally stable if it has a C’ neighborhood 9 such that every YE @ is topologically equivalent to X, i.e., there exists a homeomorphism h: M ,J mapping orbits of X onto orbits of Y and preserving their time orientation. In the boundaryless case, satisfactory sufficient conditions for structural stability have been obtained (following a conjecture of Palis-Smale [lo]) by Robbin [12] and Robinson [ 133 and recently Mafib [S] completed the proof of the necessity of these conditions for C1 structural stability. The objective of this work is to continue the line of research of [3], [4] and [6], whose final aim is to find a characterization of the structurally stable elements of 5!“‘( M, 8M). Here we shall give a complete answer to this question for vector fields whose nonwandering set is simple, i.e., consisting of a finite set of orbits. As explained in [6], our results can also be interpreted in the context of stability of Z,-equivariant vector fields. In order to make precise statements about our results let us first introduce some basic notations and definitions. If p E M is a singularity (resp. closed orbit) of X E 9” z (M, dM) we say that the weakest contraction at p is defined if among the contractive eigenvalues of DX(p) (resp. Df(p), f the Poincari map) the one with biggest real part is simple. Dually we can set the concept of when the weakest expansion at p is defined. For technical reasons we shall restrict our work to C” vector fields, and even in this space, we shall impose the generic conditions that all the singularities and periodic orbits are C* linearizable and for each critical element Q of X, the weakest contraction (resp. expansion) at 0 is defined. %c (M, 8M) denotes this open and dense subset of % m (M, Z.M). Given X ELK m (M, c?M), denote by SZ(X) the set of nonwandering points of X. Recall that if fi is a C 41 closed manifold and 9?( fi) is the set of C’ vector fields with the C’ topology, r 2 1, then a vector field X E %^‘( G) is called Morse-Smale if

A computer verification for the value of the Topological Entropy for some special subshifts in the Lexicographical Scenario

The Lorenz Attractor has been a source for many mathematical studies. Most of them deal with lower dimensional representations of its first return map. An one dimensional scenario can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics, in this case, can be modeled by a subshift in the Lexicographical model. The Lexicographical model is the set of two symbols with the topology induced by the lexicographical metric and with the lexicographical order. These subshifts are the maximal invariant set for the shift map in some interval. For some of them, the extremes of the interval are a minimal periodic sequence and a maximal periodic sequence which is an iteration of the lower extreme (by the shift map). For some of these subshifts the topological entropy is zero. In this case the dynamics (of the respective Lorenz map) is simple.Associated to any of these subshifts (let call it Λ) we consider an extension (let call it Γ) that contains Λ which also can be constructed by using an interval whose extremes can be defined by the extremes of Λ. For these extensions we present here a computer verification of the result that compute its topological entropy. As a consequence, of our results, we can say: the longer the period of the periodic sequence is then the lower complexity in the dynamics of the extension the associated map has.

Combinatorial Dynamics and an Elementary Proof of the Continuity of the Topological Entropy at θ = 101 , in the Milnor Thurston World

In the present paper we deal with the Milnor-Thurston world and we present elementary proofs of some results by combining dynamics, combinatory, linear algebra and entropy.