Equivalence Relations and Determinacy

We introduce the notion of $(Γ,E)$-determinacy for $Γ$ 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 $S=ω$. We show that for all shift actions by countable groups $G$, and any "reasonable" pointclass $Γ$, that $(Γ,E_G)$-determinacy implies $Γ$-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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