Invariant measures and equilibrium states for piecewise $C^ {1+\alpha }$ endomorphisms of the unit interval

Abstract:

A differentiable function is said to be ${C^{1 + \alpha }}$ if its derivative is a Hölder continuous function with exponent $\alpha > 0$. We show that three well-known results about invariant measures for piecewise monotonic and ${C^2}$ endomorphisms of the unit interval are in fact true for piecewise monotonic and ${C^{1 + \alpha }}$ maps. We show the existence of unique, ergodic measures equivalent to Lebesgue measure for ${C^{1 + \alpha }}$ Markov maps, extending a result of Bowen and Series for the ${C^2}$ case. We present a generalization of Adler’s Folklore Theorem for maps which satisfy a restricted mixing condition, and we show that these ${C^{1 + \alpha }}$ mixing endomorphisms possess unique equilibrium states, a result which was shown for the ${C^2}$ case by P. Walters.