Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle

TL;DR

This paper establishes the existence of Sinai-Ruelle-Bowen measures for certain noninvertible hyperbolic maps of rectangles with unbounded derivatives, extending classical one-dimensional results to more complex systems.

Contribution

It provides complete proofs for the existence of SRB measures in noninvertible, unbounded derivative systems, and develops the theory of stable and unstable manifolds in this context.

Findings

Existence of SRB measures for a class of noninvertible maps.

Extension of classical invariant measure results to systems with unbounded derivatives.

Development of stable and unstable manifold theory for these systems.

Abstract

We prove the existence of Sinai-Ruelle-Bowen measures for a class of $C^2$ self-mappings of a rectangle with unbounded derivatives. The results can be regarded as a generalization of a well-known one dimensional Folklore Theorem on the existence of absolutely continuous invariant measures. In an earlier paper analogous results were stated and the proofs were sketched for the case of invertible systems. Here we give complete proofs in the more general case of noninvertible systems, and, in particular, develop the theory of stable and unstable manifolds for maps with unbounded derivatives.