Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets

Pages: 335 - 405, Volume 20, Issue 2, February 2008      doi:10.3934/dcds.2008.20.335

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Enrique R. Pujals - IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil (email)

Abstract: Given a topologically hyperbolic attracting set of a smooth three dimensional Kupka-Smale diffeomorphism, it is proved under some dissipation hypothesis, that either the set is hyperbolic or the diffeomorphism is $C^1-$approximated by another one exhibiting either a heterodimensional cycle or a homoclinic tangency.

Keywords:  Dominated splittings, Uniformly hyperbolic systems, Bifurcations connected with nontransversal intersection, Generics dynamics.
Mathematics Subject Classification:  Primary: 37C05 , 37D05, 37G25, 37D30 ; Secondary: 37C70, 37C75.

Received: June 2006;      Revised: July 2007;      Available Online: November 2007.