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Ergodic theory and the geometry of high-dimensional metric spaces

Tim Austin: Courant Institute

Location:  Hill 705
Date & time: Friday, 03 March 2017 at 4:00PM - 4:11PM

The most basic examples of shift-systems with positive entropy are the Bernoulli shifts, under which the coordinates are independent. In the special case of Bernoulli shifts, it was shown by Ornstein that entropy is actually a complete invariant. In order to prove this, Ornstein developed a concrete necessary and sufficient condition for a general shift-system to be isomorphic to a Bernoulli shift. We also know that Bernoulli shifts often appear as images of other, more complex systems under equivariant maps: by a theorem of Sinai, this is true whenever the necessary inequality between their entropies is satisfied.

The proofs of these more advanced results requires a delicate investigation of the finite-dimensional marginals of the shift-system, regarded as a sequence of discrete probability spaces endowed with their Hamming metrics. It turns out that other ergodic theoretic consequences are related to open problems on the possible structure of such discrete `metric probability spaces'. After sketching the history of this area, I will describe some of these connections.

This talk will require a knowledge of basic real analysis and some measure theory, and some simple probability theory will be helpful. I will not assume anything from dynamics or ergodic theory.

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