Exponential volume limits

Let $M$ be a $d$-dimensional closed Riemannian manifold, let $f\in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{1+\beta }(M)$, and denote by $m$ the Riemannian volume form of $M$. We prove that if $m\circ f^{-n}\underset{n\to \mathrm{\infty }}{\overset{}{\to }}\mu$ exponentially fast, then $\mu$ is an SRB measure.