Robust Heteroclinic Tangencies
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Robust Heteroclinic Tangencies Pablo G. Barrientos1 · Sebastián A. Pérez2 Received: 23 February 2019 / Accepted: 19 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract We construct diffeomorphisms in dimension d ≥ 2 exhibiting C 1 -robust heteroclinic tangencies. Keywords Folding manifolds · Robust equidimensional tangencies · Robust heterodimensional tangencies
1 Introduction An important problem in the modern theory of Dynamical Systems is to describe diffeomorphisms whose qualitative behavior exhibits robustness under (small) perturbations and how abundant these sets of dynamics can be. Motivated by this issue, Smale (1967) introduced the hyperbolic diffeomorphisms as examples of structural stable dynamics (open sets of dynamics which are all of them conjugated). However, the transverse intersection between the invariant manifolds of basic sets was soon observed as a necessary condition (Williams 1970; Palis 1978; Mañé 1987). The main goal of this article is to study the persistence of the non-transverse intersection between those manifolds. Namely, we focus in tangencial heteroclinic orbits. A diffeomorphism f of a manifold M has a heteroclinic tangency if there are different transitive hyperbolic sets and , points P ∈ , Q ∈ and Y ∈ W s (P) ∩ W u (Q) such that def
cT = dim M − dim[TY W s (P) + TY W u (Q)] > 0 and def
dT = dim TY W s (P) ∩ TY W u (Q) > 0.
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Pablo G. Barrientos [email protected] Sebastián A. Pérez [email protected]
1
Instituto de Matemática e Estatística, UFF, Rua Prof. Marcos Waldemar de Freitas Reis, s/n, Niterói, Brazil
2
Centro de Matematica da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
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P. G. Barrientos , S. A. Pérez
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Fig. 1 Heteroclinic tangencies in dimension 3: (a) cT = 2, dT = 1 and k T = −1; (b) cT = 1, dT = 1 and k T = 0; (c) cT = 1, dT = 2 and k T = 1
The number cT is called codimension of the tangency and measures how far the tangencial intersection is from a transverse intersection. On the other hand, dT indicates the number of linearly independent common tangencial directions. Observe that cT = dT − k T
with k T = ind() − ind()
where ind() denotes the stable index of a (transitive) hyperbolic set . The integer k T is called signed co-index. Notice that when k T > 0 this number coincides with the classical co-index between and . Moreover, k T > 0 if and only if dim TY W s (P) + dim TY W u (Q) > dim M. If k T = 0, the heteroclinic tangency is called equidimensional and otherwise heterodimensional. Figure 1 illustrates the different types of heteroclinic tangencies in dimension three. Heterodimensional tangencies with signed co-index k T > 0 was introduced in Díaz et al. (2006) where interesting dynamics consequences were obtained. Indeed, the authors showed that the C 1 -unfolding of a three dimensional heterodimensional tangency (with k T = 1) leads to C 1 -robustly non-dominated dynamics and in some cases to very intermingled dynamics related to universal dynamics, for details see
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