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.