The Margulis-Ruelle inequality states that measure-theoretic entropy is controlled by Lyapunov exponents; more precisely, if $f$ is a $C^{1+\alpha}$ diffeomorphism on a $d$-dimensional manifold $M$ and $\mu$ is a Borel $f$-invariant ergodic probability measure with Lyapunov exponents $\lambda\_1, \dots, \lambda\_d$, then $h\_\mu(f) \leq \sum\_{\lambda\_i>0} \lambda\_i$.


This also holds for non-invertible $C^1$ interval maps: we have $h\_\mu(f) \leq \max(0,\lambda(\mu))$, where $\lambda(\mu) = \int \log|f'(x)|\\,d\mu(x)$. (See, for example, Proposition 4.1 of [Ledrappier, *Some properties of absolutely continuous invariant measures on an interval*, Ergodic Theory Dynam. Systems **1** (1981), 77-93].)


**Question:** Has this result been proved for *piecewise* $C^1$ interval maps? I would be surprised if it is not true in this setting, but neither I nor anyone I've asked has been able to produce a reference to a proof in the case where $f$ is a piecewise monotonic interval map without any assumption of Markov structure.



Margulis-Ruelle inequality for piecewise continuous interval maps
You can find a complete proof of a slightly more general result in my paper "A. Barrio Blaya and V. Jimenez Lopez, On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps, Discrete Contin. Dyn. Syst. 32 (2012), 433-466", see Theorem 7.1. You can download the paper from [here](http://www.um.es/docencia/vjimenez/ficheros/articulos/dcds2012.pdf).


