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Transitivity and the existence of horseshoes on the 2-torus

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Published 8 December 2022 • © 2022 IOP Publishing Ltd & London Mathematical Society
, , Citation Pollyanna Vicente Nunes and Fábio Armando Tal 2023 Nonlinearity 36 199 DOI 10.1088/1361-6544/aca252

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pollyannavnunes@gmail.com

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1 Instituto de Matemática e Estatística da Universidade de São Paulo, R. do Matão, 1010—Butantã, São Paulo 05508-090, Brazil

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2 Author to whom any correspondence should be addressed.

Dates

  1. Received 9 June 2022
  2. Accepted 11 November 2022
  3. Published

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0951-7715/36/1/199

Abstract

We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of $\mathbb{T}^2$ is homotopic to the identity and has both a fixed point and a periodic point which is not fixed, then it has a topological horseshoe. We also show that if a transitive homeomorphisms of $\mathbb{T}^2$ is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.

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10.1088/1361-6544/aca252