Some properties of singular hyperbolic attractors

Abstract

The definition of a singular hyperbolic attractor was given in a paper by Morales, Pacifico, and Pujals in 1998. A similar definition was given by Turaev and Shil'nikov in 1998. This definition was motivated on the one hand by the well-known Lorenz model, and on the other hand by the definition of a hyperbolic attractor. We prove certain properties of singular hyperbolic attractors, which are subsequently used to prove the existence of invariant measures of Sinaȋ-Bowen-Ruelle type. The main result of the paper is a proof of the existence of an invariant family of strong unstable spaces on a sufficiently representative set.Bibliography: 21 titles.

Sectional Lyapunov exponents

“…The case of Anosov flows is included in our result, considering closed manifolds instead of only compact manifolds. Moreover we also extend a result found in Sataev's paper [10].…”

“…We remark that this theorem improves a result found in [10]. First, let us recall the notation used by Sataev and set the scenario.…”

“…Corollary 2.4 (Lemma 2.12 of [10]). In this setting, the flow is sectional Anosov if and only if all singularities are hyperbolic and there exist constants C ∈ R and D > 0 such that for every orbit ϕ t (x) and every T > 0 the following inequality holds:…”

Upper Large Deviations Bound for Singular-Hyperbolic Attracting Sets

“…Comments, corollaries and possible extensions. The construction of adapted cross-sections for general singular-hyperbolic attracting sets provides an extension of the results of [18] in line with the work of [64,63]. From the representation of the global Poincaré map as a skew-product given by Theorem 2.8, we can follow [18,] to obtain Theorem 1.7.…”

Mixing and eigenfunctions of singular hyperbolic attractors

“…Семейство пространств E c x , вообще говоря, либо не продолжается до инвариантного семейства на всей окрестности U , либо продолжается неоднозначно. В [16] показано, что если x ∈ Λ, то X(x) ∈ E c…”