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Il n'y a pas de classification borélienne des homéomorphismes de Brouwer

  • FRÉDÉRIC LE ROUX (a1)
Abstract

In this paper, we consider the space {\mathcal B}_2 of the planar homeomorphisms that are obtained by gluing two translations together, endowed with the compact-open topology. We construct an embedding of the Cantor set \{0,1\}^\mathbb{Z} in {\mathcal B}_2, such that in this Cantor set of homeomorphisms the conjugacy classes are given by the orbits of the shift map on \{0,1\}^\mathbb{Z}. As a classical consequence, we explain why there cannot be any borelian classification of the conjugacy relation on the space {\mathcal B}_2.

On considère l'espace {\mathcal B}_2 des homéomorphismes du plan obtenus en recollant deux translations, muni de la topologie compacte-ouverte. On construit une famille dans {\mathcal B}_2, continûment indexée par les éléments du Cantor \{0,1\}^\mathbb{Z}, pour laquelle les classes de conjugaison correspondent aux orbites du décalage (ou “ shift ”) sur \{0,1\}^\mathbb{Z}. Ceci permet de montrer qu'il n'y a pas de classification borélienne de la relation de conjugaison sur {\mathcal B}_2.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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