  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 $+\infty>r>1$.
