Generic dichotomy for homomorphisms for $E_0^\mathbb{N}$

Assaf Shani
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Abstract

We prove the following dichotomy. Given an analytic equivalence relation $E$, either ${E_0^{\mathbb{N}}}\leq_B{E}$ or else any Borel homomorphism from $E_0^{\mathbb{N}}$ to $E$ is "very far from a reduction", specifically, it factors, on a comeager set, through the projection map $(2^{\mathbb{N}})^{\mathbb{N}}\to (2^{\mathbb{N}})^k$ for some $k\in\mathbb{N}$. As a corollary, we prove that $E_0^{\mathbb{N}}$ is a prime equivalence relation, answering a question on Clemens.
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同态$E_0^\mathbb{N}$的通用二分法
我们将证明以下二分法。给定一个解析等价关系 $E$,要么 ${E_0^{\mathbb{N}}}\leq_B{E}$ 要么任何从$E_0^{mathbb{N}}$ 到 $E$ 的伯尔同构都 "离还原很远"、具体地说,它是通过投影图$(2^{\mathbb{N}})^{mathbb{N}}\to (2^{mathbb{N}})^k$ 对某个$kin\mathbb{N}$的投影图,在一个comeager集合上形成的。作为一个推论,我们证明 $E_0^{mathbb{N}}$ 是一个首要等价关系,从而回答了一个关于克莱门斯的问题。
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