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

A Borel (Formula presented.) dynamical system (Formula presented.) is ‘almost Borel universal’ if any free Borel (Formula presented.) dynamical system (Formula presented.) of strictly lower entropy is isomorphic to a Borel subsystem of (Formula presented.), after removing a null set. We obtain and exploit a new sufficient condition for a topological (Formula presented.) 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 (Formula presented.) and dimers in (Formula presented.) are almost Borel universal.