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Dynamical Properties of Iterated Function Systems

Praveen M

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.

Dynamical Properties of Iterated Function Systems

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.

Paper Structure

This paper contains 12 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamical Systems
  4. Metric Spaces
  5. Iterated Function Systems
  6. Definitions and Major Results
  7. IFS Metric Space
  8. IFS Shift Map
  9. Periodic Points
  10. Open Set Condition
  11. Parity Evolution Operators
  12. Conclusion

Key Result

Lemma 3.1

Consider a compact metric space $(X,d)$. Let $Con(X)$ denote the collection of all contractions on $X$. Then $d_{\infty}:Con(X)\times Con(X)\to \mathbb{R}$ defined by $d_{\infty}(f,g)=\sup_{x\in X}d(f(x),g(x))$ for every $f,g \in Con(X)$, is a metric on $Con(X)$.

Theorems & Definitions (26)

  • Definition 2.1: ott2002chaos
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • proof
  • Definition 3.4
  • Theorem 3.5
  • proof
  • ...and 16 more