An inequality for the Lyapunov exponent of an ergodic invariant measure of a piecewise monotonic map of the interval.

*(English)*Zbl 0744.58042
Lyapunov exponents, Proc. Conf., Oberwolfach/Ger. 1990, Lect. Notes Math. 1486, 227-231 (1991).

[For the entire collection see Zbl 0733.00015.]

Two often used measures of chaos are entropy \(h_ \mu\) and the Lyapunov exponent \(\lambda_ \mu\). If \(\mu\) is an invariant ergodic probability measure both are defined by ergodic averages almost everywhere. The paper proves that \(h_ \mu\leq \lambda_ \mu\) for a class of piecewise monotone interval maps, which allow infinitely many pieces with certain restrictions, and are differentiable Lebesgue-a.e. with a weak condition on their variation. This constitutes an improvement of the result by F. Ledrappier [Ergodic Theory Dyn. Syst. 1, 77-93 (1981; Zbl 0487.28015)] where stronger conditions of differentiability were imposed and only finitely many pieces were allowed.

Two often used measures of chaos are entropy \(h_ \mu\) and the Lyapunov exponent \(\lambda_ \mu\). If \(\mu\) is an invariant ergodic probability measure both are defined by ergodic averages almost everywhere. The paper proves that \(h_ \mu\leq \lambda_ \mu\) for a class of piecewise monotone interval maps, which allow infinitely many pieces with certain restrictions, and are differentiable Lebesgue-a.e. with a weak condition on their variation. This constitutes an improvement of the result by F. Ledrappier [Ergodic Theory Dyn. Syst. 1, 77-93 (1981; Zbl 0487.28015)] where stronger conditions of differentiability were imposed and only finitely many pieces were allowed.

Reviewer: G.Swiatek (Stony Brook)

##### MSC:

37A99 | Ergodic theory |