The Hutchinson-Barnsley theory for iterated function systems with general measures
Elismar R. Oliveira, Rafael R. Souza
TL;DR
The paper extends Hutchinson–Barnsley theory to Iterated Function Systems with Measures (IFSm) where the index set Λ is compact and the map-selection probabilities $q_x$ depend on the current state $x$. It defines the transfer operator $B_q$ and the Markov operator $T_q$, proves the existence and uniqueness of a fractal attractor $A_{\mathcal{R}}$ and an invariant Hutchinson measure $\mu_{\mathcal{R}}$, and shows that $\mathrm{supp}(\mu_{\mathcal{R}})=A_{\mathcal{R}}$ with convergence of iterates in the Hausdorff and Monge–Kantorovich metrics. The work also establishes stochastic stability: $\mu_{\mathcal{R}}$ depends continuously on the family of state-dependent measures $q_x$ in the MK topology, under suitable regularity conditions. A variety of examples—including constant probabilities, Lipschitz potential representations, mixtures, and GIFS-based constructions—demonstrate the framework’s breadth and potential links to thermodynamic formalism. The results broaden the applicability of fractal and invariant-measure theory to nonuniform, state-dependent random dynamics, with implications for stochastic stability and geometric structure of attractors.
Abstract
In this work we present iterated function systems with general measures(IFSm) formed by a set of maps $τ_λ$ acting over a compact space $X$, for a compact space of indices, $Λ$. The Markov process $Z_k$ associated to the IFS iteration is defined using a general family of probabilities measures $q_x$ on $Λ$, where $x \in X$: $Z_{k+1}$ is given by $τ_λ(Z_k)$, with $λ$ 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 $q_x$.