The Maximal Entropy Measure of Fatou Boundaries

TL;DR

This paper investigates the properties of the maximal entropy measure on Fatou boundaries of rational maps, establishing conditions under which at most one Fatou component has a boundary with positive measure, extending to geometrically finite maps.

Contribution

It introduces new results on the maximal entropy measure of Fatou boundaries, generalizing hyperbolicity to geometric finiteness for rational maps.

Findings

At most one Fatou boundary has positive MME measure under specified conditions.

The results apply to maps with infinitely many Fatou components and either disconnected Julia set or hyperbolicity.

Extension of hyperbolic results to geometrically finite rational maps.

Abstract

We look at the maximal entropy (MME) measure of the boundaries of connected components of the Fatou set of a rational map of degree greater than or equal to 2. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.