Foundations of local iterated function systems

TL;DR

This paper systematically studies local iterated function systems, proving their attractors' properties, exploring their combinatorial structures, and illustrating their diversity beyond classical models.

Contribution

It introduces a comprehensive framework for local IFSs, including attractor existence, code space construction, and examples of non-classical behaviors.

Findings

Local IFSs admit compact attractors.

Under contractivity, code spaces and extended shifts describe admissible compositions.

Examples include non-self-similar attractors and systems not modeled by subshifts of finite type.

Abstract

In this paper we present a systematic study of continuous local iterated function systems. We prove local iterated function systems admit compact attractors and, under a contractivity assumption, construct their code space and present an extended shift that describes admissible compositions. In particular, the possible combinatorial structure of a local iterated function system is in bijection with the space of invariant subsets of the full shift. Nevertheless, these objects reveal a degree of unexpectedness relative to the classical framework, as we build examples of local iterated function systems which are not modeled by subshifts of finite type and give rise to non self-similar attractors. We also prove that all attractors of graph-directed IFSs are obtained from local IFSs on an enriched compact metric space. We provide several classes of examples illustrating the scope of our results, emphasizing both their contrasts and connections with the classical theory of iterated function systems.