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Foundations of local iterated function systems

Elismar R. Oliveira, Paulo Varandas

TL;DR

This work develops a general framework for local iterated function systems (local IFSs) on a compact metric space, where domain restrictions $X_j$ render the classical Hutchinson–Barnsley theory inapplicable. It establishes the existence of a compact local attractor $A_{ rak X}$ and a hierarchy of basins, and it shows how, in the contractive setting, the local dynamics admit a symbolic model via a semiconjugacy to a shift on an invariant code space. The authors construct generalized and extended shift spaces to encode admissible compositions and infinite orbits, relate local attractors to graph-directed IFSs through enriched spaces, and prove exponential convergence of essential iterates to $A_{ rak X}$. Together, these results reveal rich, sometimes non-self-similar attractors and provide tools for analyzing their combinatorial, dynamical, and topological structure. The work thus broadens the standard IFS theory to encompass local constraints and their intricate symbolic dynamics, with clear consequences for both theory and potential applications.

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.

Foundations of local iterated function systems

TL;DR

This work develops a general framework for local iterated function systems (local IFSs) on a compact metric space, where domain restrictions render the classical Hutchinson–Barnsley theory inapplicable. It establishes the existence of a compact local attractor and a hierarchy of basins, and it shows how, in the contractive setting, the local dynamics admit a symbolic model via a semiconjugacy to a shift on an invariant code space. The authors construct generalized and extended shift spaces to encode admissible compositions and infinite orbits, relate local attractors to graph-directed IFSs through enriched spaces, and prove exponential convergence of essential iterates to . Together, these results reveal rich, sometimes non-self-similar attractors and provide tools for analyzing their combinatorial, dynamical, and topological structure. The work thus broadens the standard IFS theory to encompass local constraints and their intricate symbolic dynamics, with clear consequences for both theory and potential applications.

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.
Paper Structure (19 sections, 19 theorems, 134 equations, 2 figures)

This paper contains 19 sections, 19 theorems, 134 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Local iterated function systems
  4. Topological aspects
  5. Code space for contractive local iterated function systems
  6. Generalized shift spaces
  7. Existence of a local attractor and its topological basin
  8. Proof of Theorem \ref{['thm:topological']}
  9. Code spaces
  10. Code space for contractive IFS
  11. Code space for contractive local IFSs
  12. Proof of Theorem \ref{['thm:classification']}
  13. Dynamics of contractive local IFSs
  14. Orbits of a contractive local IFS
  15. Proof of Theorem \ref{['thmC']}
  16. ...and 4 more sections

Key Result

Theorem 1

Let $R_{\mathfrak X}=(X_j, f_{j})_{1\leqslant j \leqslant n}$ be a local IFS with local attractor $A_{\mathfrak X}$. The following properties hold:

Figures (2)

  • Figure 1.1: Local attractor $A_{\mathfrak X}$ that is non-transitive and not self-similar, in which the endpoints are marked in grey and $A^\infty_{\mathfrak X}$ is marked in black (cf. Example \ref{['ex:MapleSierpinski']})
  • Figure 8.1: Iterates $F_{\mathfrak X}^1(X),F_{\mathfrak X}^2(X), F_{\mathfrak X}^5(X)$ and $F_{\mathfrak X}^{15}(X)$ (left to right, up to bottom).

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 1
  • Remark 2.5
  • Definition 2.6
  • Theorem 2
  • Theorem 3
  • Remark 2.7
  • ...and 45 more