Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond
Nishant Chandgotia, Tom Meyerovitch · Proceedings of the London Mathematical Society · 2021
A Borel Z d dynamical system ( X , S ) is ‘almost Borel universal’ if any free Borel Z d 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 Z d 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 Z d and dimers in Z 2 are almost Borel universal.