Typical path components in tent map inverse limits

Philip Boyland, André de Carvalho, Toby Hall Fundamenta Mathematicae MSC: Primary 37B45; Secondary 37E05. DOI: 10.4064/fm810-1-2020 Published online: 20 March 2020

Abstract

In the inverse limit $\hat {I}_s$ of a tent map $f_s$ restricted to its core, the set ${\mathcal {GR}}$ of points whose path components are bi-infinite and bi-dense has full measure with respect to the measure induced on $\hat {I}_s$ by the unique absolutely continuous invariant measure of $f_s$. With respect to topology, there is a dichotomy. When the parameter $s$ is such that the critical orbit of $f_s$ is not dense, ${\mathcal {GR}}$ contains a dense $G_\delta $ set. In contrast, when the critical orbit of $f_s$ is dense, the complement of ${\mathcal {GR}}$ contains a dense $G_\delta $ set.

Authors

  • Philip BoylandDepartment of Mathematics
    University of Florida
    372 Little Hall
    Gainesville, FL 32611-8105, U.S.A.
    e-mail
  • André de CarvalhoDepartamento de Matemática Aplicada
    IME-USP
    Rua do Matão 1010
    Cidade Universitária
    05508-090 São Paulo SP, Brazil
    e-mail
  • Toby HallDepartment of Mathematical Sciences
    University of Liverpool
    Liverpool L69 7ZL, UK
    e-mail

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