Partially hyperbolic and transitive dynamics generated by heteroclinic cycles. (English) Zbl 0972.37018

Díaz, Lorenzo J.; Rocha, Jorge

Ergodic Theory Dyn. Syst. 21, No. 1, 25-76 (2001).

The authors study \(C^k\) diffeomorphisms, \(k \geq 1\), \(f :M \rightarrow M\), where \(M\) is a compact and smooth boundaryless manifold, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). They prove that if \(f\) can not be \(C^k\) approximated by diffeomorphisms with homoclinic tangencies, then \(f\) is in the closure of an open set \({\mathcal U} \subset \text{Diff}^k(M)\) consisting of diffeomorphisms \(g\) with a non-hyperbolic transitive set \(\Lambda_g\) which is locally maximal and strongly partially hyperbolic. As a consequence, in the case of 3-dimensional manifolds, they give new examples of open sets of \(C^1\) diffeomorphisms for which residually infinitely sinks and sources coexist. They also prove that the occurence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.

Reviewer: Jerzy Ombach (Kraków)

MSC:

37D30	Partially hyperbolic systems and dominated splittings
37C29	Homoclinic and heteroclinic orbits for dynamical systems

Keywords:

homoclinic orbits; heteroclinic orbits; homoclinic tangencies; partially hyperbolic sets