SRB measures for C $\infty$ surface diffeomorphisms

TL;DR

This paper characterizes the existence of SRB measures for smooth surface diffeomorphisms based on the measure of points with positive Lyapunov exponents, and describes the basins of these measures.

Contribution

It provides a precise criterion for the existence of SRB measures in terms of Lyapunov exponents and Lebesgue measure, extending results to various smoothness classes.

Findings

SRB measures exist iff the set of points with positive Lyapunov exponent has positive Lebesgue measure.

Basins of SRB measures cover this set Lebesgue almost everywhere.

Results apply to $C^ ext{infinity}$ and $C^r$ surface diffeomorphisms with $r>1$.

Abstract

A $C^\infty$ surface diffeomorphism admits a SRB measure if and only if the set \left \{x, \limsup_n \frac{1}{ n} \log \|d_xf^n \|> 0\right\} has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for $C^r$ surface diffeomorphisms with $+\infty>r>1$.