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Some aspects of rotation theory on compact abelian groups

Volume 161 / 2020

Manuel Cruz-López, Francisco J. López-Hernández, Alberto Verjovsky Colloquium Mathematicum 161 (2020), 131-155 MSC: Primary 22-XX, 37-XX; Secondary 22Cxx, 37Axx, 37Bxx. DOI: 10.4064/cm7593-12-2018 Published online: 24 March 2020

Abstract

We present a generalization of Poincaré’s rotation theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, i.e., groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements à la Poincaré and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between rotation theory and entropy for these homeomorphisms, since unlike the case of the circle, for the solenoids considered here there are homeomorphisms (not homotopic to the identity) with positive entropy.

Authors

  • Manuel Cruz-LópezDepartamento de Matemáticas
    Universidad de Guanajuato
    Jalisco s/n
    Mineral de Valenciana
    Guanajuato, Gto. 36240, Mexico
    e-mail
  • Francisco J. López-HernándezInstituto de Física
    Universidad Autónoma de San Luis Potosí
    Av. Manuel Nava No. 6
    Zona Universitaria
    San Luis Potosí, SLP 78290, Mexico
    e-mail
  • Alberto VerjovskyInstituto de Matemáticas
    Unidad Cuernavaca
    Universidad Nacional Autónoma de México
    Apdo. Postal 2
    Cuernavaca, Mor. 2000, Mexico
    e-mail

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