The discrete charm of iterated function systems. A computer scientist's perspective on approximation of IFS invariant sets and measures

TL;DR

This paper explores the approximation of invariant sets and measures generated by discrete iterated function systems on countable grids, focusing on the applicability of the random iteration algorithm in computational models.

Contribution

It introduces a framework for analyzing discrete IFSs on countable spaces and examines the effectiveness of the random iteration algorithm for their approximation.

Findings

Random iteration algorithm can approximate discrete IFS invariant sets.

Discretization of hyperbolic IFSs is a special case within this framework.

Discrete spaces model numerical computation environments.

Abstract

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.