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Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions

100%
Díaz L., Gelfert K.
Fundamenta Mathematicae
|
2012
|
tom 216
|
nr 1
55-100
EN
We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of $F|_{Λ}$ contains a gap and hence gives rise to a first order phase transition. A major part of the proofs relies on the analysis of an associated iterated function system that is genuinely non-contracting.
2
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Generic diffeomorphisms on compact surfaces

81%
Abdenur F., Bonatti C., Crovisier S., Díaz L.
Fundamenta Mathematicae
|
2005
|
tom 187
|
nr 2
127-159
EN
We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion, we obtain some related results about C¹-generic properties of surface diffeomorphisms involving homoclinic classes, chain-recurrence classes, and hyperbolicity. In particular, it is shown that on a connected surface the C¹-generic diffeomorphisms whose non-wandering sets have non-empty interior are the Anosov diffeomorphisms.
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