Analogues of Khintchine's theorem for random attractors

TL;DR

This paper establishes conditions under which Khintchine-like theorems apply to random iterated function systems, extending classical Diophantine approximation results to stochastic self-similar and self-affine fractals.

Contribution

It provides the first sufficient conditions for Khintchine-type theorems to hold in the context of random fractals generated by iterated function systems.

Findings

Khintchine-like theorems hold almost surely for certain random IFS

Conditions identified for stochastic self-similar systems

Extension of classical Diophantine approximation results to random fractals

Abstract

In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.