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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.

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See <a… 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? – Anthony Quas Nov 6 '12 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. – Tom Kempton Nov 7 '12 at 13:51
up vote 5 down vote accepted

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).



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Thanks Matheus, i think I'll have to spend some time with this book! – Tom Kempton Nov 7 '12 at 13:45
This book was indeed what I needed. If anyone else reading this needs the same results and can't get their hands on the book then the results from the crucial chapter can also be found in Ledrappier and Strelcyn, A proof of the estimation from below in Pesin's entropy formula, ETDS 1982. – Tom Kempton Nov 21 '12 at 14:11

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