SRB measures for \(C^{\infty }\) surface diffeomorphisms

A $C^{\mathrm{\infty }}$ $C^{\infty }$ smooth surface diffeomorphism admits an SRB measure if and only if the set $\{x,{lim\hspace{0.17em}sup}_n\frac{1}{n}log\parallel d_x{f}^n\parallel >0\}$ $\{ x, \ \limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}\|>0\}$ 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 $C^r$ $C^{r}$ surface diffeomorphisms with $+\mathrm{\infty }>r>1$ $+\infty >r>1$ .