Counting Problem for Some Random Conformal Iterated Function Systems

TL;DR

This paper investigates the counting problem in random conformal iterated function systems, revealing that growth rates differ significantly from deterministic systems due to the influence of recurrent random walk behavior.

Contribution

It introduces a novel analysis of counting in random dynamical systems, highlighting the impact of recurrent behavior on growth rates.

Findings

Counting growth is non-exponential on full measure sets.

Recurrent behavior of random walks significantly influences counting.

Differences between random and deterministic systems are characterized.

Abstract

This paper studies the counting problem in random dynamical systems. We noticed that the nature of counting in the random setting is completely different than that of the deterministic systems in the sense that non-exponential growth is constructed on a set of full measure. The recurrent behavior of random walks plays a major role in counting in the random setting.