SRB measures for C $\mathrm{\infty }$ surface diffeomorphisms

A $C^{\mathrm{\infty }}$ 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 $+\mathrm{\infty }>r>1$.