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Continuous 2-colorings and topological dynamics

Tom 586 / 2023

Dominique Lecomte Dissertationes Mathematicae 586 (2023), 1-92 MSC: Primary 03E15; Secondary 54H05, 37B05, 37B10. DOI: 10.4064/dm870-7-2023 Opublikowany online: 10 August 2023

Streszczenie

We first consider the class $\mathfrak K$ of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for $\mathfrak K$ made up of countable graphs, comparing them with the quasi-order $\preceq^i_c$ associated with injective continuous homomorphisms. We prove that the size of such a basis is sharp, using odometers. However, using odometers again, we prove that there is no antichain basis in $\mathfrak K$, and provide infinite descending chains in $\mathfrak K$. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of $2^\omega$ is Borel reducible to the equivalence relation associated with $\preceq^i_c$. We also prove that there is no antichain basis in the class of graphs on a zero-dimensional Polish space with continuous chromatic number at least three. We study the graphs induced by a continuous function, and show that any $\preceq^i_c$-basis for the class of graphs induced by a homeomorphism of a zero-dimensional metrizable compact space with continuous chromatic number at least three must have size continuum, using odometers or subshifts.

Autorzy

  • Dominique LecomteSorbonne Université, CNRS, IMJ-PRG
    75005 Paris, France
    and
    Université de Paris, IMJ-PRG
    75013 Paris, France
    and
    Université de Picardie, I.U.T. de l’Oise, site de Creil
    60100 Creil, France
    e-mail

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