Strange Attractors with One Direction of Instability

Toward a theory of rank one attractors

“…They include (1) a bound on the number of ergodic SRB measures, (2) conditions that imply ergodicity and mixing for SRB measures, (3) almost-everywhere behavior in the basin, (4) statistical properties of SRB measures such as correlation decay and CLT, and (5) coding of orbits on the attractor, growth of periodic points, etc. A 2D version of these results is published in [WY1]. Additional work is needed in higher dimensions due to the increased complexity in geometry.…”

“…Our treatment of the subject is necessarily more conceptual as we replace the equation of the Hénon maps by geometric conditions. A 2D version of these results was published in [WY1].…”

“…A 2D version of monotone branches was introduced in [WY1]. They were not used, however, in the inductive construction of the dynamical picture.…”

Markov extensions and conditionally invariant measures for certain logistic maps with small holes

“…Lemma 2.5 of [WY1] implies that maps satisfying the Misiurewicz condition belong to M. In viewing this class of maps, we divide the phase space into two parts: (−δ 0 , δ 0 ) and its complement. Part (b) of the definition says that f is essentially expanding outside of (−δ 0 , δ 0 ) while part (c) ensures that when orbits come close to the critical point, they subsequently spend enough time away from (−δ 0 , δ 0 ) for their derivatives to recover some exponential growth.…”

Hyperbolic times: frequency versus integrability

“…In this direction we mention [7,9,12] for quadratic maps, [8,10,16,18,20] for Hénon-like diffeomorphisms, and [1,5,19] for a generalized higher dimensional version of the quadratic and Hénon-like maps. The dynamics of all these systems is characterized by the existence of regions of the phase space where the system displays some hyperbolicity, together with critical regions where some strong non-hyperbolic behavior appears.…”