A topological and dynamic approach to fractal geometry

Author(s)

Gabriela Cristina da Silva

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2023-03-06

Examining board members

Paolo Piccione; Fernando Manfio; Francesco Mercuri

• Advisor: Paolo Piccione

Abstract

Mostly, fractals are defined as attractors of an Iterated Function System (IFS). Defining fractals in this way often facilitates the calculation of their Hausdorff dimension, since doing the calculation by definition is, in general, complicated. The main objective of this master thesis is to clearly present a proof of Moran\'s theorem --- which guarantees us that, if F is the attractor of an IFS whose contractions are similarities that satisfy the Set Condition Open (OSC), then the similarity dimension of F coincides with its Hausdorff dimension. The present work is a contribution to the study of Fractal Geometry from a topological and dynamic point of view. Although Fractal Geometry and Chaotic Dynamics are traditionally studied independently, in 2014 Barnsley showed the presence of chaos in fractals. In this work, we present a result that relates chaotic dynamics and fractals. More specifically, we prove that the change transformation associated with a totally disconnected IFS composed of two or more transformations is chaotic according to Devaney\'s definition. (AU)

• FAPESP's process: 20/13512-3 - A topological and dynamical approach to Fractal Geometry
• Grantee: Gabriela Cristina da Silva
• Support Opportunities: Scholarships in Brazil - Master