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Article Dans Une Revue Dissertationes Mathematicae Année : 2023

Continuous 2-colorings and topological dynamics

Résumé

We first consider the class 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 k made up of countable graphs, comparing them with the quasi-order <=(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 k, and provide infinite descending chains in 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 <=(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 <=(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.

Dates et versions

hal-04428476 , version 1 (31-01-2024)

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Dominique Lecomte. Continuous 2-colorings and topological dynamics. Dissertationes Mathematicae, 2023, 586, pp.1-92. ⟨10.4064/dm870-7-2023⟩. ⟨hal-04428476⟩
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