On the convergence of sequences in the space of $n$-iterated function systems with applications

TL;DR

This paper investigates the convergence properties of sequences of n-iterated function systems within a metric space, providing theoretical results and exploring practical applications in fractal construction.

Contribution

It introduces a new metric on the space of n-iterated function systems and analyzes convergence, Cauchy, and decreasing sequences within this framework.

Findings

Sequences of n-iterated function systems can be convergent under certain conditions.

The paper establishes criteria for decreasing and Cauchy sequences of iterated function systems.

Practical applications in fractal generation and related fields are discussed.

Abstract

This article discusses the notion of convergence of sequences of iterated function systems. The technique of iterated function systems is one of the several methods to construct objects with fractal nature, and the fractals obtained with this method are mostly self-similar. The progress in the theory of fractals has found potential applications in the fields of physical science, computer science, and economics in abundance. This paper considers the metric space of $n$- iterated function systems by introducing a metric function on the set of all iterated function systems on a complete metric space consisting of $n$ contraction functions. Further, sequences of $n$- iterated function systems with decreasing, eventually decreasing, Cauchy and convergent properties are discussed. Some results on sequences of $n$- iterated function systems and sequences of contractions are obtained. The practical usage of the theory discussed in the article is explored towards the end.