Caltech/UCLA Joint Analysis Seminar
This talk is about the structure theory of measure-preserving systems: transformations of a finite measure space that preserve the measure. Many important examples arise from stationary processes in probability, and simplest among these are the i.i.d. processes. In ergodic theory, i.i.d. processes are called Bernoulli shifts. Some of the main results of ergodic theory concern an invariant of systems called their entropy, which turns out to be intimately related to the existence of 'structure preserving' maps from a general system to Bernoulli shifts.
I will give an overview of this area and its history, ending with a recent advance in this direction. A measure-preserving system has the weak Pinsker property if it can be split, in a natural sense, into a direct product of a Bernoulli shift and a system of arbitrarily low entropy. The recent result is that all ergodic measure-preserving systems have this property. Its proof depends on a new theorem in discrete probability: a probability measure on a finite product space such as A^n can be decomposed as a mixture of a controlled number of other measures, most of them exhibiting a strong 'concentration' property. I will sketch the connection between these results and the proof of the latter, to the extent that time allows.