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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.

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    $\begingroup$ Which assumption of Ruelle's theorem do you want to drop? $\endgroup$ Apr 15 '19 at 11:38
  • $\begingroup$ I want to drop the regularity of $f$. $\endgroup$
    – M.H
    Apr 15 '19 at 18:13
  • $\begingroup$ If there is no derivative, how do you DEFINE the characteristic exponents? $\endgroup$ Apr 15 '19 at 22:38
  • $\begingroup$ 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 $\endgroup$
    – M.H
    Apr 16 '19 at 6:04
  • $\begingroup$ What do you mean regularity of f? $\endgroup$
    – Adam
    May 28 '19 at 16:25

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