The size of Julia sets of quasiregular maps

TL;DR

This paper investigates the Hausdorff dimension of Julia sets for quasiregular maps, revealing they can have zero dimension and analyzing their measure properties with respect to specific gauge functions.

Contribution

It extends Fatou-Julia theory to quasiregular maps, showing the possibility of zero-dimensional Julia sets and establishing measure properties with new gauge functions.

Findings

Julia sets can have Hausdorff dimension zero.

Existence of a gauge function with positive, finite Hausdorff measure.

Extension of classical iteration theory to quasiregular maps.

Abstract

Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context, in contrast to the case of rational functions, the Julia set may have Hausdorff dimension zero. On the other hand, we exhibit a gauge function depending on the degree and the dilatation such that the Hausdorff measure with respect to this gauge function is always positive, but may be finite.