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

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

Revised after Referee's comments; 70 pages