Rigidity in mildly smooth 1-D systems

• Grant number: 17/50139-6
• Support Opportunities: Regular Research Grants
• Start date: December 01, 2017
• End date: November 30, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Agreement: Imperial College, UK
• Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
• Principal Investigator: Edson de Faria
• Grantee: Edson de Faria
• Principal researcher abroad: Sebastian Van Strien
• Institution abroad: Imperial College London, England
• 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

The aim of this proposal is to foster existing research collaborations between a group of researchers at Imperial College and IME-USP. The dynamical systems groups at Imperial College and IME-São Paulo are world-Leading, in particular in the field of one-dimensional dynamics. Some of the most exciting and promising developments in this field are aimed at understanding universal structures and geometric rigidity: universal scaling both in parameter and in phase space that occur at arbitrary deep levels. These scaling are independent of the particular system considered. (Such results often go under the name of "Feigenbaum-Coullet-Tresser" renormalization.) Unfortunately, so far it has not been possible to develop satisfactory technology to obtain such results in the natural setting of smooth systems (rather than in the much more restrictive setting of real analytic maps). Recent results on quasi-symmetric rigidity for smooth maps form the starting point for a theory that should make it possible to treat smooth cases more easily. The proposed collaboration creates a platform to do exactly this; the expertise of the two teams form a perfect fit to tackle these questions. (AU)