TITLE: Pesin Entropy Formula
QUESTION [10 upvotes]: 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.

REPLY [7 votes]: Dear Tom,
I believe that the Pesin entropy formula for maps with singularities (such as piecewise smooth maps) is discussed in the book "Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities" of A. Katok and J.-M. Strelcyn (see its parts III and IV). 
Best, 
Matheus