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.

positiveLyapunov exponents? – Anthony Quas Nov 6 '12 at 11:42