Degenerations of Complex Dynamical Systems

TL;DR

This paper investigates the limits of maximal measures in degenerating sequences of rational maps, revealing they become atomic, and introduces a new measure quantization technique on the Berkovich projective line.

Contribution

It provides a novel analysis of measure limits in degenerating rational maps and introduces a new quantization method on the Berkovich line.

Findings

Limit measures are countable sums of atoms.

Unique limiting measure exists for degenerating families.

New quantization technique for measures on Berkovich line.

Abstract

We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere must be a countable sum of atoms. For a 1-parameter family f_t of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on the Riemann sphere as the family degenerates. The family f_t may be viewed as a single rational function on the Berkovich projective line over the completion of the field of formal Puiseux series in t, and the limiting measure on the Riemann sphere is the "residual measure" associated to the equilibrium measure on the Berkovich line. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure there.