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.

  • 3
    $\begingroup$ 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? $\endgroup$ Commented Nov 6, 2012 at 11:42
  • $\begingroup$ Thanks Anthony, that's a useful link. And yes, $\Sigma(x)$ is the sum of positive Lyapunov exponents, I'll edit to include this. $\endgroup$ Commented Nov 7, 2012 at 13:51

1 Answer 1


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).



  • $\begingroup$ Thanks Matheus, i think I'll have to spend some time with this book! $\endgroup$ Commented Nov 7, 2012 at 13:45
  • $\begingroup$ 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. $\endgroup$ Commented Nov 21, 2012 at 14:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.