Equivalence Relations and Determinacy

We introduce the notion of $(\mathrm{\Gamma },E)$-determinacy for $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$, that $(\mathrm{\Gamma },E_G)$-determinacy implies $\mathrm{\Gamma }$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^{\mathbb{Z}}$.