Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties

Abstract

We introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

A piecewise hyperbolic map with absolutely continuous invariant measure

“…The fat baker's transformations are a special case of the Belykh map, introduced in [3] by Belykh. In [4], Schmeling and Troubetzkoy extended Pesin's results from [5] to non-invertible maps. They gave a condition for Young's dimension formula to hold and studied the Belykh map for a wider range of parameters as an example of their results.…”

“…The map f satisfies the conditions (H1), (H5)-(H9) in [5] and this implies that there exists C, q > 0 such that ðf Àn UðS, "ÞÞ C" q for all n > 0, where UðS, "Þ denotes the "-neighbourhood of S. This implies that SBR ðÃÞ ¼ SBR ðÃÞ ¼ 1.…”

“…This set has positive Lebesgue measure if is taken sufficiently small, see [5]. It is even true that…”

On the Structure of Generalized Hyperbolic Attractors of Mappings That Are Not One-to-One

“…Since the first part of the assertion of Corollary 1 is a straightforward corollary to Theorem 1 and Λ is an attractor by definition, it remains to show that Λ is hyperbolic. By [1,2], to this end, in some neighborhood U ⊂ IN of the set Λ, it is necessary to find (or construct) two families of sectors C u ((x, y)) and C s ((x, y)) with the following properties:…”

“…It is also obvious that F is the simplest representative of such mappings. 1 Before continuing the exposition of the results, we note that "unimodular mappings of the unit square" include sets of isomorphisms whose properties (for λ ∼ = 1 and for sufficiently small s > 0) are similar to those of the Henon diffeomorphism (x, y) → (1 − ax 2 + y, bx) [10] and the Lozi homeomorphism (x, y) → (1 − a|x| + y, bx) [1] (with a ∼ = 2 and sufficiently small |b| > 0). This permits one to consider the above-mentioned mappings as a possible counterpart of the Lozi and Henon mappings in the class of non-one-to-one mappings.…”