# Combinatorial invariants of metric filtrations and automorphisms; the universal adic graph

## TL;DR

This paper introduces a combinatorial classification of metric filtrations using a measure on hierarchies, leading to new invariants for automorphisms, and constructs a universal graph with an adic structure to realize all automorphisms.

## Contribution

It provides a complete combinatorial invariant for metric filtrations and constructs a universal adic graph capable of representing all automorphisms.

## Key findings

- Complete invariant of metric filtrations via combinatorial schemes
- Introduction of a universal adic graph for automorphisms
- New metric invariants derived from combinatorial schemes

## Abstract

We suggest a combinatorial classification of metric filtrations in measure spaces; a complete invariant of such a filtration is its combinatorial scheme, a measure on the space of hierarchies of the group~$\mathbb Z$. In turn, the notion of combinatorial scheme is a source of new metric invariants of automorphisms approximated via basic filtrations. We construct a universal graph endowed with an adic structure such that every automorphism can be realized in its path space.