Looking for a counterexample for Ruelle's inequality on compact manifold

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.

• Which assumption of Ruelle's theorem do you want to drop? Apr 15 '19 at 11:38
• I want to drop the regularity of $f$.
– M.H
Apr 15 '19 at 18:13
• If there is no derivative, how do you DEFINE the characteristic exponents? Apr 15 '19 at 22:38
• the point is, the measure of The singular set S contains those points $x$ where $f$ is either not defined, is discontinuous,not differentiable or else $D f(x)$ is non-invertible, and we have same inequality for the systems with non-flat singularities with two excess condition's, for more details look at this paper arxiv.org/pdf/math/0601449.pdf
– M.H
Apr 16 '19 at 6:04
• What do you mean regularity of f?