Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps

TL;DR

This paper establishes the existence of SRB measures for certain dissipative maps with non-uniform hyperbolicity, including systems lacking dominated splitting, expanding the class of systems where statistical properties can be rigorously analyzed.

Contribution

It introduces new methods to prove SRB measures for systems without dominated splitting, broadening understanding of hyperbolic dynamics in dissipative maps.

Findings

SRB measures exist for a class of dissipative maps with effective hyperbolicity

Systems without dominated splitting can have SRB measures

Provides examples of such systems with positive volume initial conditions

Abstract

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.