Random complex dynamics and semigroups of holomorphic maps

TL;DR

This paper explores the complex behavior of random rational map dynamics on the Riemann sphere, revealing phenomena like singular functions and chaos suppression due to generator cooperation, with implications for ergodic and potential theory.

Contribution

It provides a systematic analysis of random complex dynamics and semigroup behaviors, introducing new phenomena not seen in traditional single-map iteration.

Findings

Random dynamics often suppress chaos compared to individual maps.

Singular functions analogous to devil's staircase appear in the limit.

Functions representing escape probabilities are continuous and vary on Julia sets.

Abstract

We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, "singular functions on the complex plane" appear. In particular, we consider the functions $T$ which represent the probability of tending to infinity with respect to the random dynamics of polynomials. Under certain conditions these functions $T$ are complex analogues of the devil's staircase and Lebesgue's singular functions. More precisely, we show that these functions $T$ are continuous on the Riemann sphere and vary only on the Julia sets of associated semigroups. Furthermore, by using ergodic theory and potential theory, we investigate the non-differentiability and regularity of these functions. We find many phenomena which can hold in the random complex dynamics and the dynamics of semigroups of rational maps, but cannot hold in the usual iteration dynamics of a single holomorphic map. We carry out a systematic study of these phenomena and their mechanisms.