New criteria for hyperbolicity based on periodic sets

Abstract

We prove some criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a C 1 -open set U then there exists an open and dense subset A ⊂ U of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms.

On critical orbits and sectional hyperbolicity of the nonwandering set for flows

“…Moreover, using suspensions, our theorem implies a result found in [11]. First, we recall his notion of nonuniformly hyperbolic sets for diffeomorphisms.…”

“…Actually, we also have another reformulation of our result in terms of residual sets. Furthermore, using suspensions we reobtain the analogous result in [11], thus a residual subset of Axiom A diffeomorphisms.…”

“…In this note, we extend the definition of nonuniformly hyperbolic set, introduced in [11], to the sectional hyperbolic context. Namely, we define nonuniformly sectional hyperbolic sets.…”

“…Namely, we define nonuniformly sectional hyperbolic sets. Using the ideas and some results from [5] and the techniques from [11], we prove that if an open subset U of the space X 1 (M) of C 1 -vector fields contains a residual subset S whose elements have a nonuniformly sectional hyperbolic critical set, then U has a residual subset whose elements are vector fields with sectional hyperbolic nonwandering set. Actually, we also have another reformulation of our result in terms of residual sets.…”

Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles

“…In the case of the C 1 -topology, one can describe the mechanisms to deduce simple Lyapunov spectrum for all invariant measures assuming the same property at periodic points, in the spirit of the previous works [3,6,7,14]. On the one hand we prove that, if all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum and the same property holds in a C 1 neighborhood of f (c.f.…”