Statistical and non-statistical properties of dynamical systems

• Grant number: 22/16259-2
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: July 01, 2023
• End date: June 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Ali Tahzibi
• Grantee: Douglas Coates
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 17/06463-3 - Probabilistic and algebraic aspects of smooth dynamical systems, AP.TEM

Abstract

Physical measures are an important tool for studying the long-term behaviour of chaotic dynamical systems, however, their existence is not always guaranteed. In fact there are many open problems and conjectures about the existence and non-existence of these measures, including the well known conjectures of J. Palis (which asserts the existence of physical measures generically) and of M. Viana (which asserts that if the Lyapunov exponents of almost every point are non-zero then a physical measure exists). This proposal outlines novel methods for studying dynamical systems which lack physical measures. We are principally interested in the existence of families that exhibit non-statistical behaviour; whether this behaviour can be robust to perturbation; and what, if any, meaningful descriptions can be obtained for the long term behaviour of non-statistical systems. (AU)