Canonization of smooth equivalence relations on infinite-dimensional perfect cubes

TL;DR

This paper introduces a canonization scheme for smooth equivalence relations on infinite-dimensional perfect cubes, revealing a structural dichotomy on infinite perfect products that simplifies their classification.

Contribution

It provides a new method to canonize smooth equivalence relations on $ eals^ abla$ modulo restriction to infinite perfect products, establishing a clear structural dichotomy.

Findings

Existence of an infinite perfect product where one relation is contained in the other

A structural form where equivalence implies equality at a specific coordinate

A form where coordinate-wise agreement implies relation equivalence

Abstract

A canonization scheme for smooth equivalence relations on $\mathbb R^\omega$ modulo restriction to infinite perfect products is proposed. It shows that given a pair of Borel smooth equivalence relations $\mathsf E,\mathsf F$ on $\mathbb R^\omega$, there is an infinite perfect product $P\subseteq\mathbb R^\omega$ such that either ${\mathsf F}\subseteq{\mathsf E}$ on $P$, or, for some $j<\omega$, the following is true for all $x,y\in P$: $x\,\mathsf E \,y$ implies $x(j)=y(j)$, and $x\restriction{(\omega\smallsetminus\{j\})}=y\restriction{(\omega\smallsetminus\{j\})}$ implies $x\,\mathsf F \,y$.