Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

Details Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

2010-04-30

arxiv.org/abs/1004.5453v1

Mathematics

Dynamical Systems

Scientific paper

In the present paper, we consider a special set of maps (Benedicks-Carleson toy models) acting on a two dimensional rectangle. For this special type of systems, we show that, if one deals in $C^2$-topology, there are open set of diffeomorphisms which are not hyperbolic, while in the $C^1$-topology, the Axiom A property is open and dense. In particular, our results support Smale's conjecture saying that Axiom A is a $C^1$-open and dense property among $C^1$-diffeomorphisms of compact surfaces.

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