Random compositions of $T^2$ homeomorphisms and rotation sets

• Grant number: 23/16187-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: July 01, 2024
• End date: June 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: André Salles de Carvalho
• Grantee: Catalina Freijo
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 16/25053-8 - Dynamics and geometry in low dimensions, AP.TEM

Abstract

This project studies random compositions of homeomorphisms of surfaces, particularly of homeomorphisms of $\T^2$ in the isotopy class of the identity. While the usual techniques for assessing complexity of dynamical behaviour of  cocycles of diffeomorphisms is the study of Lyapunov exponents, we intend to apply Brouwer theory  and rotation theory like techniques in our study. Our main goal is to provide a solid foundation for defining rotation sets for cocycles instead of single maps, and to study how much of the theory for a single dynamic carries over to our setting. Several relevant related questions are proposed, and we also describe our initial results in the field.