Universal models in ergodic theory

Abstract:
A topological dynamical system is a pair (X,T) where T is a homeomorphism
of a compact space X. A measure preserving action is a triple (Y, \mu, S)
where Y is a standard Borel space, \mu is a probability measure on X and S
is a measurable automorphism of Y which preserves the measure \mu. We say
that (X,T) is universal if it can embed any measure preserving action
(under some suitable restrictions).

Krieger’s generator theorem shows that if X is A^Z (bi-infinite sequences
in elements of A) and T is the transformation on X which shifts its
elements one unit to the left then (X,T) is universal. Along with Tom
Meyerovitch, we establish very general conditions under which Z^d (where
now we have d commuting transformations on X)-dynamical systems are
universal. These conditions are general enough to prove that

1) A self-homeomorphism with non uniform specification on a compact metric
space (answering a question by Quas and Soo and recovering recent results
by David Burguet)
2) A generic (in the sense of dense G_\delta) self-homeomorphism of the
2-torus preserving Lebesgue measure (extending result by Lind and
Thouvenot to infinite entropy)
3) Proper colourings of the Z^d lattice with more than two colours and the
domino tilings of the Z^2 lattice (answering a question by Şahin and
Robinson)

are universal. Our results also extend to the almost Borel category giving
partial answers to some questions by Gao and Jackson.

The talk will not assume background in ergodic theory and dynamical systems.