Topological horseshoes for transitive 2-torus homeomorphisms

• Ano de defesa: 2021
• Autor(a) principal: Nunes, Pollyanna Vicente
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Dehn twist; Ferradura topológica; Homeomorfismo; Homeomorphism; Isotopic to identity; Isotópico à identidade; Topological horseshoe; Toro; Torus
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45132/tde-30062021-104510/

Based on rotation theory and forcing theory for transverse trajectories of surface homeomorphisms, in this work we study the relation between transitive homeomorphism of the 2-torus and the existence of a topological horseshoe. Let f be a transitive homeomorphism of the 2-torus. In the case where f is isotopic to the identity, we show that f has a topological horseshoe, if f has a fixed point and a non-fixed periodic point. In the case where a power f^k, k > 1, is isotopic to identity but f itself is not, we show that if f has at least one fixed point and has no topological horseshoe then the rotation set of some lift of f to the plane is only the origin. We also study the case where a power f^k, k 1, of f is isotopic to Dehn twist. In this case we show that f has a topological horseshoe, if f has at least a fixed point.