Does someone have a reference where the Ruelle inequality would be proved in the following context.

Let $M$ be a non compact smooth manifold, and $f:M\to M$ be a $C^1$-diffeomorphism (or $C^2$, or smooth), whose differential is uniformly bounded ($\sup_{x\in M}\|T_xf\|<\infty$) on $M$.

Assume maybe that $M$ satisfies an additional assumption : [???? to complete ??]

Let $\mu$ be a $f$-invariant probability measure on $M$. Then $$ h_\mu(f)\le \int_M \sum_{i:\chi_i(f)>0} \chi_i(x)\dim E_i(x) d\mu(x) $$ where the numbers $\chi_i(x)$ are the Lyapounov exponents, and $E_i(x)$ the corresponding spaces in the Oseledets decomposition.