Entropy estimates and a stronger theorem on the existence of horseshoes for a forcing theory for surface homeomorphism

Author(s)

Everton Juliano da Silva

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2019-06-17

Examining board members

Fabio Armando Tal; Philip Lewis Boyland; André Salles de Carvalho; Alejandro Kocsard; Alejandro Miguel Passeggi Diaz Robles

• Advisor: Fabio Armando Tal

Abstract

In this work we study the minimum topological entropy value for one class of maps isotopics to the identity in oriented surfaces (without border, not necessary compacts and possibly of finite type) under the point of view strictly topological. This study is done using the new forcing theory to transverse trajectories from Le Calvez and Tal which it is based to equivariant Brouwer Theory, on what it is possible to leaf surfaces with leaves related to plane Brouwer theory. The main result in this work is a improvement in the estimates from the topological entropy obtained by Le Calvez and Tal in one recent work where the authors seek topological horseshoes on oriented surfaces using tools very similar to that are shown here. One application of the above result is done using maps on S^2 that have a fixed point whose trajectory by the isotopy of this point do not be homotopic to a multiple of a simple loop. With these hypotheses, we improve the estimates given by Le Calvez and Tal on what is found a strictly positive minimum value to the topological entropy of this map. (AU)

• FAPESP's process: 14/12872-5 - Transitive and entropy for homeomorphisms of surface
• Grantee: Everton Juliano da Silva
• Support Opportunities: Scholarships in Brazil - Doctorate