Stability of the maximal measure for piecewise monotonic interval maps

Abstract

Let $T:X\to{\Bbb R}$ be a piecewise monotonic map, where $X$ is a finite union of closed intervals. Define $R(T)=\bigcap_{n=0}^{\infty} \overline{T^{-n}X}$, and suppose that $(R(T),T)$ has a unique maximal measure $\mu$. The influence of small perturbations of $T$ on the maximal measure is investigated. If $(R(T),T)$ has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map $\tilde{T}$, which is contained in a sufficiently small neighbourhood of $T$, has a unique maximal measure $\tilde{\mu}$, and the map $\tilde{T}\mapsto\tilde{\mu}$ is continuous at $T$.