# Pesin Entropy Formula

In the form that I've seen it stated, the Pesin entropy formula states that if $M$ is a compact Riemannian manifold and $f$ is a $C^{1+\alpha}$ diffeomorphism of $M$ that preserves smooth invariant measure $\mu$, then

$$h_{\mu}(f)=\int_M \Sigma(x)d\mu(x)$$

where $\Sigma(x)$ denotes the sum of the positive lyapunov exponents of $f$ at $x$.

$Question:$ Does the above hold if $f$ is only piecewise $C^{1+\alpha}$?

In fact I'm really interested in a specific example called the random $\beta$-transformation, which is interesting in the study of Bernoulli convolutions and $\beta$-expansions.

This can be written as a map on $[0,1]^2$ which is piecewise linear (on four pieces) but not Markov in general. It preserves a measure $\mu$ equivalent to Lebesgue measure. I'd be really grateful to hear of a reference where the Pesin entropy formula has been pushed forward to this kind of situation.

• See <a href=mathoverflow.net/questions/79800/… question</a> for something closely related (but in one dimension). Also, you presumably mean that $\Sigma(x)$ is the sum of the positive Lyapunov exponents? Nov 6, 2012 at 11:42
• Thanks Anthony, that's a useful link. And yes, $\Sigma(x)$ is the sum of positive Lyapunov exponents, I'll edit to include this. Nov 7, 2012 at 13:51