Stochastic Equidistribution and Generalized Adelic Measures

TL;DR

This paper introduces generalized adelic measures to study the dynamics of stochastic rational maps on the projective line, extending previous concepts to a broader setting and proving new equidistribution results.

Contribution

It generalizes adelic measures to handle infinite families of maps over algebraic closures, enabling new equidistribution theorems in stochastic dynamics.

Findings

Proves an equidistribution theorem for generalized adelic measures.

Establishes equidistribution for random backwards orbits in stochastic dynamics.

Extends the framework of adelic measures to infinite and non-field-defined families.

Abstract

We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rationals, using a framework established by Allcock and Vaaler. This turns our heights from sums over places into integrals. We prove an equidistribution result for generalized adelic measures, and use this result to prove an equidistribution result for random backwards orbits in stochastic arithmetic dynamics.