A two-dimensional version of the folklore theorem

Abstract

We formulate some su cient conditions for the existence of Sinai-Ruelle-Bowen measures for piecewise C 2 di eomorphisms with unbounded derivatives. The result can be viewed as a two-dimensional version of the well known one-dimensional Folklore Theorem on the existence of absolutely continuous invariant measures. Here we formulate the results and outline the main ideas and tools of our approach. The detailed version will appear elsewhere.

Thermodynamic formalism for some systems with countable Markov structures

“…For the proof of the Folklore theorem , the ergodic properties of µ and the history of the question see for example [3] and [15]. In [8] , [9] the Folklore Theorem was generalized to two-dimensional maps F which piecewise coincide with certain hyperbolic diffeomorphisms f i . As in the one-dimensional situation there is an essential difference between a finite and an infinite number of f i .…”

“…2 Model under consideration. Geometric and hyperbolicity conditions 1. As in [8] , [9] we consider the following 2-d model. Let Q be the unit square.…”

“…Here we use hyperbolicity conditions from [9]. In [8] we used hyperolicity conditions from [4] which implied the invariance of cones and uniform expansion with respect to the sum norm…”

“…Let us denote µ S the invariant measure on Λ constructed in [8], [9] following Sinai method, and let µ RB be the invariant measure on Λ constructed above following Ruelle-Bowen method, see [11], [6]. Let µ 1 be the projection of µ S onto one-sided sequences, and let µ be the measure on one-sided sequences constructed above by Ruelle-Bowen method.…”

Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps

“…In this paper, we study a class of dynamical systems having the most extended structure called generalized horseshoe maps. The generalized horseshoe map initially defined by Jakobson and Newhouse in [4] to detect the SRB measure in the most general case. The generalized horseshoe map is a piecewise hyperbolic map defined on a countable family of vertical strips.…”