Equilibrium measures for uniformly quasiregular dynamics

Abstract

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism f of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure µf , which is balanced and invariant under f and non-atomic, and whose support agrees with the Julia set of f. Furthermore we show that f is strongly mixing with respect to the measure µf . We also characterize the measure µf using an approximation property by iterated pullbacks of points under f up to a set of exceptional initial points of Hausdorff dimension at most n − 1. These dynamical mixing and approximation results are reminiscent of the Mattila-Rickman equidistribution theorem for quasiregular mappings. Our methods are based on the existence of an invariant measurable conformal structure due to Iwaniec and Martin and the A-harmonic potential theory.