SRB measures for C $\infty$ surface diffeomorphisms

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 $+\infty >r>1$.

References in corpus (7)

• Takens' last problem and existence of non-trivial wandering domains
• Uniqueness of SRB measures for transitive diffeomorphisms on surfaces
• The geometric approach for constructing Sinai-Ruelle-Bowen measures
• Continuity properties of Lyapunov exponents for surface diffeomorphisms
• C^r surface diffeomorphisms with no maximal entropy measure
• Dirac physical measures on saddle-type fixed points
• Another proof of Burguet's existence theorem for SRB measures of $C^{\infty }$ surface diffeomorphisms