Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces

TL;DR

This paper introduces a class of three-dimensional piecewise hyperbolic maps with two directions of instability, establishing the existence of Sinai-Ruelle-Bowen measures and illustrating results with computer simulations.

Contribution

It extends the theory of SRB measures to 3D piecewise hyperbolic maps with two unstable directions, providing new existence results and computational examples.

Findings

Existence of SRB measures for the class of maps studied.

Illustrative computer simulations confirming theoretical results.

Extension of hyperbolic dynamics theory to higher dimensions.

Abstract

A class of piecewise $C^2$ Lozi-like maps in three-dimensional Euclidean spaces is introduced, and the existence of Sinai-Ruelle-Bowen measures is studied, where the dimension of the instability is equal to two. Further, an example with computer simulations is provided to illustrate the theoretical results.