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.



Pesin Entropy Formula
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](http://www.ams.org/mathscinet-getitem?mr=872698)" of A. Katok and J.-M. Strelcyn (see its parts III and IV). 


Best, 


Matheus


