Let $M$ be a compact differentiable manifold, and $f:M\to M$ be a $C^1$- smooth diffeomorphism.

If Assume that $\mu$ be a $f$-invariant probability measure on $M$. Then D.Ruelle proved that $$ h_\mu(f)\le \int_M \sum_{i:\chi_i(f)>0} \chi_i(x)\dim E_i(x) d\mu(x) $$ where $h_\mu(f)$ is measure theoretical entropy, $\chi_i(x)$ are the Lyapounov exponents and $E_i(x)$ the corresponding Lyapunov spaces.

I'm looking for an example on compact manifold that above Ruelle's inequality fails for it.