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On universal continuous actions on the Cantor set
Dynamical Systems · Submitted 14 Mar 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$.
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The abstract, sentence by sentence
Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$. We extend our theorem for (nonfree) Borel $(\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Using the notion of proper Cantor colorings we prove the following theorem. | method 0.85 | 0.85 | 0.03 | asserted 0.95 |
| s1 | For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$. | result 0.96 | 0.93 | 0.94 | asserted 1.00 |
| s2 | We extend our theorem for (nonfree) Borel $(\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup. | result 0.52 | 0.94 | 0.08 | asserted 1.00 |
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Original abstract
Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$. We extend our theorem for (nonfree) Borel $(\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup.
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The result will follow easily from the following lemma, which is an extended version of the result in [ref] , who dealt with the case of free actions.
Realizations of countable Borel equivalence relations · Compact action realizations · 1.00
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