Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

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