The Hutchinson-Barnsley theory for iterated function systems with general measures

TL;DR

This paper extends the Hutchinson-Barnsley theory to iterated function systems with general measures, establishing the existence of attractors and invariant measures for a broad class of stochastic processes.

Contribution

It introduces a generalized framework for IFS with measures depending on the state, proving key properties like attractor existence and measure stability.

Findings

Existence of topological attractor for the IFS

Existence of invariant attracting measure

Support of the measure coincides with the attractor

Abstract

In this work we present iterated function systems with general measures(IFSm) formed by a set of maps $\tau_{\lambda}$ acting over a compact space $X$, for a compact space of indices, $\Lambda$. The Markov process $Z_k$ associated to the IFS iteration is defined using a general family of probabilities measures $q_x$ on $\Lambda$, where $x \in X$: $Z_{k+1}$ is given by $\tau_{\lambda}(Z_k)$, with $\lambda$ randomly chosen according to $q_x$. We prove the existence of the topological attractor and the existence of the invariant attracting measure for the Markov Process. We also prove that the support of the invariant measure is given by the attractor and results on the stochastic stability of the invariant measures, with respect to changes in the family $q_x$.