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Fixed points for branched covering maps of the plane

Volume 254 / 2021

Alejo García Sassi Fundamenta Mathematicae 254 (2021), 1-14 MSC: Primary 37B20; Secondary 37E30. DOI: 10.4064/fm765-4-2020 Published online: 10 December 2020

Abstract

A well-known result of Brouwer states that any orientation preserving homeomorphism of the plane with no fixed points has an empty non-wandering set. In particular, the existence of an invariant compact set implies the existence of a fixed point. In this paper we give sufficient conditions for degree 2 branched covering maps of the plane to have a fixed point, namely:
$\bullet$ A totally invariant compact subset that does not separate the critical point from its image.
$\bullet$ An invariant compact subset with a connected neighbourhood $B$ such that $\mathrm {Fill}(B \cup f(B))$ does not contain the critical point nor its image.
$\bullet$ An invariant continuum such that the critical point and its image belong to the same connected component of its complement.

Authors

  • Alejo García SassiInstituto de Matemática y Estadística
    Facultad de Ingeniería
    Universidad de la República
    Julio Herrera y Reissig 565
    11300 Montevideo, Uruguay
    e-mail

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