On universal continuous actions on the Cantor set

2018/03/14 by Gábor Elek, Elek, Gábor
Mathematics · #46L55 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1803.05461

openalex publication_date 2018/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group Γ, there exists a free continuous action ζ: Γ\curvearrowright C on the Cantor set, which is universal in the following sense: for any free Borel action α: Γ\curvearrowright X on the standard Borel space, there exists an injective Borel map Θ_α: X→ C such that Θ_α∘ α=ζ∘ Θ_α. We extend our theorem for (nonfree) Borel (Γ,Z)-actions, where Z is a uniformly recurrent subgroup.

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• Amenable purely infinite actions on the non-compact Cantor set
• Universal properties of group actions on locally compact spaces
• Nilpotent Cantor actions
• Limit group invariants for non-free Cantor actions