Stability Theory for Local Iterated Function Systems

TL;DR

This paper develops a stability theory for contractive local iterated function systems on compact metric spaces, highlighting how contraction and combinatorial rigidity influence stability and instability.

Contribution

It introduces new stability criteria for local IFSs, including conditions for topological stability and examples of instability derived from beta-transformations.

Findings

Concordant shadowing implies upper semicontinuity of the local attractor.

Under the open set condition, a strong form of topological stability is established.

Contractive graph-directed IFSs are shown to be topologically stable.

Abstract

We develop a stability theory for contractive local IFSs on compact metric spaces. Unlike the classical global setting, local systems may exhibit a richer symbolic and geometric structure, including code spaces that are not of finite type and attractors with endpoints, leading to new mechanisms of instability. We first prove that concordant shadowing implies upper semicontinuity of the local attractor and persistence of the code space, yielding a criterion for combinatorial stability under perturbations. Under the open set condition, we establish a strong form of topological stability for combinatorially stable contractive local systems, and prove the converse implication on compact manifolds of dimension at least three. In particular, we show that contractive graph-directed IFSs are topologically stable. We also construct contractive local IFSs derived from beta-transformations that are combinatorially unstable. These results show that stability in the local setting is governed by the interplay between contraction and the combinatorial rigidity of the code space. Applications to graph-directed IFSs and pseudogroup actions are also given.