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 µ 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 µ f . We also characterize the measure µ 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 A-harmonic potential theory.where Df is the point-wise operator norm of the differential Df of the mapping f and J f is the Jacobian determinant of the differential. A quasiregular endomorphism f : M → M is said to be uniformly quasiregular if there exists K 1 so that all the iterates of f are K-quasiregular.Results of Peltonen [21] and Astola, Kangaslampi and Peltonen [1] show that the existence of branching uniformly quasiregular dynamics is not limited to S n . However, results of Martin, Meyer and Peltonen [16] show that, among space forms, only spherical space forms admit branching uniformly quasiregular endomorphisms. Furthermore, by results of Bridson, Hinkkanen and Martin [4, Corollary 5.3], all quasiregular endomorphisms of closed n-manifolds having a torsion-free, non-elementary, and word-hyperbolic fundamental group, are in fact quasiconformal. We refer the reader to [4,16] for the terminology and related results. Since the

Entropy in uniformly quasiregular dynamics

“…In the proof of Theorem 1.1 we obtain estimates h( f ) ≥ log deg f and h( f ) ≤ log deg f for the entropy by different methods. The lower bound employs Lyubich's variational method [19] and the properties [16,27] of the equilibrium measure µ f associated f . The upper bound is related to [8, (5.0)] in Gromov's article and it follows from isoperimetric arguments for Federer-Fleming currents [6].…”

“…In order to obtain the lower bound h( f ) ≥ log deg f , the main obstacle is the lack of continuity of the derivative D f . For this, we use the f -balanced measure µ f from [27] and the integer valued cochain x → i(x, f ) given by the local index of the map f in place of cochain x → J f (x), which is only measurable in this setting.…”

“…Let f : M → M be a uniformly quasiregular self-map of degree at least 2 on a closed, connected, and oriented Riemannian n-manifold M, which is not a rational cohomology sphere. Then the measure µ f from [27] is a measure of maximal entropy.…”

“…The Julia set J ( f ) is non-empty if deg f > 1. In this case, there exists by [27] an fbalanced probability measure µ f on M, that is,…”

Uniform cohomological expansion of uniformly quasiregular mappings

“…However, we have not found it discussed in the literature; see Heinonen–Kilpeläinen–Martio [, pp. 263–268], or for example Okuyama–Pankka , for the push‐forward of functions.…”