# Borel Subsystems and Ergodic Universality for Compact Zd-systems via Specification and Beyond

Tom Meyerovitch, Ben-Gurion University of the Negev
Tuesday, March 12, 2019 - 1:30pm
PDL C-401

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 particularalmost 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.

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