# Borel subsystems and ergodic universality for compact $\mathbb Z^d$-systems via specification and beyond

## TL;DR

The paper introduces a new criterion for topological systems to be almost Borel universal, enabling them to model all lower-entropy free Borel systems, with applications to generic homeomorphisms, specification, and combinatorial models.

## Contribution

It establishes a sufficient condition for almost Borel universality and applies it to various systems, including generic homeomorphisms and combinatorial models in lattice systems.

## Key findings

- Generic homeomorphisms model any ergodic transformation.
- Non-uniform specification implies almost Borel universality.
- Certain lattice colorings and dimers are almost Borel universal.

## Abstract

A Borel system $(X,S)$ is `almost Borel universal' if any free Borel dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient condition for a topological dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a `generic' homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non-uniform specification implies almost Borel universality, and that $3$-colorings in $\mathbb Z^d$ and dimers in $\mathbb Z^2$ are almost Borel universal