On a universal Borel adic space

TL;DR

This paper demonstrates that the uniadic graph and its adic automorphism serve as a universal model for all aperiodic Borel automorphisms, establishing a foundational link in Borel dynamics.

Contribution

It proves the universality of the uniadic graph and adic automorphism for aperiodic Borel automorphisms, extending the understanding of Borel automorphism classification.

Findings

Uniadic graph automorphism is Borel universal.

Every aperiodic Borel automorphism is isomorphic to a restriction of the uniadic automorphism.

Development of basic filtrations and combinatorial definiteness concepts.

Abstract

We prove that the so-called uniadic graph and its adic automorphism are Borel universal, i.e., every aperiodic Borel automorphism is isomorphic to the restriction of this automorphism to a subset invariant under the adic transformation, the isomorphism being defined on a universal (with respect to the measure) set. We develop the concept of basic filtrations and combinatorial definiteness of automorphisms suggested in our previous paper.