A Borel system (X,T) is ``almost Borel universal'' if any free Borel dynamical system (Y,S) of strictly lower entropy is isomorphic to a Borel subsystem of (X,T), after removing a null set. In particular``almost Borel universality'' implies ``ergodic universality'' which means that the ergodic measures for the system contain an isomorphic copy of any free measure preserving action of entropy strictly lower than the topological entropy. Krieger's generator theorem states that the full shift is universal in the ergodic sense (more generally any topologically mixing shift of finite type). Mike Hochman proved it is ``almost Borel universal’'. In this talk I will describe a new sufficient condition for a topological dynamical system to be almost Borel universal.
and explain how we use our main result (combined with previously known results) to deduce various conclusions and answer a number of questions. For instance:
- A ``generic'' homeomorphisms of a compact manifold of topological dimension at least two can model any ergodic transformation,
- Non-uniform specification implies almost Borel universality
- 3-colorings in ^d and dimers in ^2 are almost Borel universal.