Orbital rigidity and flexibility in low-dimensional systems

• Grant number: 25/13366-0
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2025
• End date: August 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Edson de Faria
• Grantee: Miguel Ratis Laude
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/07076-4 - Dynamics and geometry in low dimensions, AP.TEM

Abstract

An aspect of the small-scale geometry of a differentiable dynamical system that has been little studied is that of the orbital rigidity/flexibility of such a system. The objective of this project is to study this problem in the context of diffeomorphisms of the two-dimensional torus. More precisely, the objective is to find necessary and sufficient conditions for a diffeomorphism of the torus to be orbitally rigid. This problem is connected to the more classical problem of determining the centralizer of a diffeomorphism of the torus, that is, the group of homeomorphisms of the torus that commute with the given diffeomorphism. A specific question in this context is to obtain necessary and sufficient conditions for the centralizer to be contained in the group of quasi-conformal homeomorphisms of the torus.