Dynamical Properties of Iterated Function Systems

TL;DR

This paper introduces a dynamical systems framework for iterated function systems (IFS), enabling rigorous analysis of their evolution over time and opening new avenues for applications in various scientific fields.

Contribution

It develops a complete metric space of IFS and defines an evolution operator, establishing a foundation for analyzing IFS dynamics rigorously.

Findings

Proves the continuity of the IFS shift map.

Characterizes periodic points of the shift map.

Links parity evolution operators to IFS similarity dimensions.

Abstract

Iterated function systems (IFS) provide a powerful method for constructing fractals and modeling complex structures. This paper develops the notion of a dynamical system of IFS to study how an initial IFS evolves over time. We construct a complete metric space consisting of countable IFS as the state space. An evolution operator is defined that maps the state space across time while satisfying properties of a dynamical system. We analyze the resulting IFS dynamics, introducing concepts like the IFS shift map and parity evolution operators. The shift map is proven continuous and its periodic points characterized. Results relate properties of parity evolution operators to similarity dimensions of the resulting IFS. Overall, this dynamical system framework allows rigorous analysis of IFS behavior over time. By providing tools to model fractal progression, it has significant applications in areas including image analysis, biology, and physics. The dynamics and properties established lay groundwork for future studies and practical models leveraging IFS.