Compactifications and measures for rational maps

TL;DR

This paper extends the measure of maximal entropy to compactified parameter and moduli spaces of rational maps, resolving discontinuities and answering a question by DeMarco.

Contribution

It introduces new compactifications that allow continuous extension of the measure of maximal entropy for rational maps.

Findings

Measure extends continuously to the resolution space of parameter space.

Measure extends continuously to the resolution space of moduli space.

Provides a description of limiting dynamics for certain sequences.

Abstract

We study extensions of the measure of maximal entropy to suitable compactifications of the parameter space and the moduli space of rational maps acting on the Riemann sphere. For parameter space, we consider a space which resolves the discontinuity of the iterate map. We show that the measure of maximal entropy extends continuously to this resolution space. For moduli space, we consider a space which resolves the discontinuity of the iterate map acting on its geometric invariant theory compactification. We show that the measure of maximal entropy, barycentered and modulo rotations, also extends continuously to this resolution space. Thus, answering in the positive a question raised by DeMarco. A main ingredient is a description of limiting dynamics for some sequences.