Entropy of discretized Anosov flows

• Grant number: 23/00676-6
• Support Opportunities: Regular Research Grants
• Start date: January 01, 2024
• End date: December 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
• Principal Investigator: Ali Tahzibi
• Grantee: Ali Tahzibi
• Principal researcher abroad: Sylvain Crovisier
• Institution abroad: Centre National de la Recherche Scientifique , France
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• City of the host institution: São Carlos
• Associated researchers: Gabriel Ponce ; José Régis Azevedo Varão Filho
• Associated research grant: 17/06463-3 - Probabilistic and algebraic aspects of smooth dynamical systems, AP.TEM

Abstract

This projects intends to describe some classes of differentiable dynamics. Uniformly hyperbolic systems have been well understood and the efforts nowfocus on partially hyperbolic systems: the complications come from the presenceof a center direction. Many previous works have analyzed the compact centercase, and we propose to deepen the study of Discretized Anosov Flows (DAF), which include deformations of the time-one map of Anosov flows, and of somegeodesic flows. We would like to take advantage from the recent progresses inthe compact center case and tackle difficulties of the non-compact center case. The idea of this project emerged after some members of our team have obtained two independent results for DAF. One will further our analysis by describing (1) the measures maximizing the entropy (geometrical properties, finiteness, ergodicproperties, center Lyapunov exponents), (2) the Margulis families of measureswhich also capture the properties of the invariant foliations (AU)