Stability properties in dynamical systems

• Grant number: 13/15022-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2013
• End date: July 31, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Fábio Armando Tal
• Grantee: Mateus Ferraz Dias
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 11/16265-8 - Low dimensional dynamics, AP.TEM

Abstract

The main goal of this research project is to expose the student to modern techniques and problems in dynamical systems, with emphasis in surface dynamics and rotation vectors for surface homeomorphisms.We expect the candidate, at the completion of the project, to possess the necessary knowledge to further pursue this line of research on a graduate course, and to be familiar with most basic concepts in the field. With this in mind, we will begin by studying the dynamics of interval mappings and of circle homeomorphisms, as a medium to explain the fundamental theory of modern dynamical systems, and then proceed to the study of circle endomorphisms and related questions and structures, such as degree and rotation set of an endomorphism.The basic concepts will be presented focusing on the structural stability and hiperbolicity of invariant sets, both for isolated systems and one and two parameters families.