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

Abstract

We consider a piecewise monotonic and piecewise continuous map T on the interval. Under a weak condition on the derivative of T, we show for an ergodic invariant probability measure μ that hμ≤max{0, λμ}, where hμ denotes the entropy and λμ the Ljapunov exponent of μ.