Ergodic Theory and Dynamical Systems

Research Article

Equilibrium states for piecewise monotonic transformations

Franz Hofbauera1 and Gerhard Kellera2

a1 Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien

a2 Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-6900 Heidelberg

Abstract

We show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:

(i) sup φ — inf φ <htop(T) and φ is of bounded variation.

(ii) φ satisfies a variation condition and T has a local specification property.

(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

(Received February 01 1982)