Number of Equilibrium States of Piecewise Monotonic Maps of the Interval

Abstract

We prove a bound of the form suggested by S. Newhouse for the number of measures with maximal entropy for a piecewise monotonic map with N monotonicity intervals: $4(N - 1)$. More generally we consider a potential $\phi$ of bounded distortion. If $\sup \phi < P(f,\phi )$, we give an explicit bound in terms of N and of the pressure.