Closing Lemma: a study of the art and the proof of the case C^1

• Grant number: 24/01771-5
• Support Opportunities: Scholarships in Brazil - Master
• Start date: May 01, 2024
• End date: February 28, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Tiago de Carvalho
• Grantee: Tácito de Deus Ferreira Soares
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• Associated research grant: 22/02819-6 - Intermittent vector fields: theoretical aspects and applications, AP.PNGP.PI

Abstract

We will address one of the most challenging problems in the dynamical systems theory which is called "Closing Lemma". The problem statement is as follows:Consider a non-periodic point x0 such that the trajectory through x0 returns to a neighborhood ofx0 infinite times. There is a small disturbation of the original system so that the newsystem has a closed orbit passing through x0 ?This problem appears, as Problem 10, on the list of main mathematical challengesfor the 21st century prepared by Smale. There are several versions of this problem in the literature and several interesting results.In the specific case of smooth vector fields, a perturbation C0 of the original field triviallycan generate a closed orbit, but extend this class of differentiability to superior orders is an extremely complex job and is still open. Throughout this project we will review the literature on the topic and give the proof for case C^1.