Equilibrium measures for uniformly quasiregular dynamics

TL;DR

This paper proves the existence and key properties of an equilibrium measure for uniformly quasiregular maps on manifolds, showing it is invariant, supports the Julia set, and the map is strongly mixing with respect to it.

Contribution

It establishes the existence, invariance, and mixing properties of the equilibrium measure for uniformly quasiregular dynamics, extending understanding of their ergodic behavior.

Findings

Existence of a unique equilibrium measure $$ for uniformly quasiregular maps.

The measure $$ is invariant, balanced, and supported on the Julia set.

The map is strongly mixing with respect to $$.

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 $\mu_f$, 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 $\mu_f$. We also characterize the measure $\mu_f$ 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 $\cA$-harmonic potential theory.