Iterated Function System: Construction of Fractals

• Grant number: 26/01965-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: March 01, 2026
• End date: February 28, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Tiago Pereira da Silva
• Grantee: Igor Campos de Castro
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 23/13706-0 - EPSRC-FAPESP predicting critical transitions in complex dynamical networks: reduction and learning, AP.TEM

Abstract

The study of random dynamical systems offers crucial tools to model various kinds of phenomena that incorporate stochasticity, such as natural dynamics including Earth's climate. In this context, this project aims to study the construction of random dynamical systems generated by a finite set of contractions on a complete metric space and their relation to fractals. Particular attention is devoted to the exploration of both local and global evolution of a system in the setting where different probabilities are associated with each of the transformations.Firstly, the formal study of iterated function systems will be developed via the Barnsley-Hutchinson operator, defined on the space of compact subsets endowed with the Hausdorff distance. In this stage, the fundamental elements underlying the construction of this operator, as well as its contractive properties, will be investigated. Subsequently, the random iteration of the transformations via independent and identically distributed random variables will be introduced, leading to the construction of coding maps and the analysis of the almost sure asymptotic behavior of the orbits toward the Hutchinson attractor.Finally, these results will be applied to the development of Barnsley's chaos game, exploring the generation of fractals and their use in methods of image compression. (AU)