Homoclinic Tangencies and Hyperbolicity for Surface Diffeomorphisms

Abstract

We prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms, the ones exhibiting a homoclinic tangency are C 1 dense. This represents a step towards the global understanding of dynamics of surface diffeomorphisms.

Another dichotomy for surface diffeomorphisms

“…Indeed, if such a tangency exists, then we could unfold it to obtain g close to g (thus g ∈ U(f )) exhibiting p ∈ Saddle(g ) such that ∠(E s p , E u p ) < γ, contradicting Proposition 2.2. Then, by the main theorem in [7], there is a dense subset A(f ) ⊂ U(f ) all of whose elements are Axiom A without cycles (see p. 966 of [7]). Then, A(f ) is also open since the set of Axiom A diffeomorphisms without cycles is (e.g.…”

“…The proof is based on Franks's Lemma and the main theorem in [7]. Let us state a corollary of Theorem 1.1.…”

Hamiltonian suspension of perturbed Poincaré sections and an application

“…[14,15]). Later, Pujals and Sambarino [16] proved this conjecture for the C 1 topology in the context of diffeomorphisms on compact surfaces. Notice that there are no heterodimensional cycles for surface diffeomorphisms.…”

Entropy-Expansiveness and Domination for Surface Diffeomorphisms

“…In case of ω(x) being included in a periodic circle C this circle is normally hyperbolic attracting a neighborhood V of C and points in V converge exponentially fast to C. If f is C 2 then as in [33] we conclude that the dynamics by f τ (τ being the period of C) in C is conjugate to an irrational rotation while if f is just C 1 we only have semi-conjugacy. (We may have a Cantor set as Ω(f/C) in C and wandering intervals.)…”

“…In any case we may assume that for all point z ∈ I we have that W cs loc (z) is a stable manifold (see [33,Corollary 3.3]) and so W 2 . But this is just used when (f n (I)) → 0 in order to argue as in Schwartz's proof of the Denjoy property ( [36]).…”