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arXiv · 2003.02238

Equivalence Relations and Determinacy

Logan Crone·Lior Fishman·Stephen Jackson
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Abstract

We introduce the notion of $(\Gamma,E)$-determinacy for $\Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$ or $S=\omega$. We show that for all shift actions by countable groups $G$, and any "reasonable" pointclass $\Gamma$, that $(\Gamma,E_G)$-determinacy implies $\Gamma$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^\mathbb{Z}$.

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Logan Crone, Lior Fishman, Stephen Jackson. 2020-03-04. Equivalence Relations and Determinacy. https://arxiv.org/abs/2003.02238

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