### Ergodic theory and strong randomness notions

**Franklin, J ***(University of Connecticut)*

Thursday 05 July 2012, 14:30-15:00

Seminar Room 1, Newton Institute

#### Abstract

There has been a great deal of interest recently in the connection between
algorithmic randomness and ergodic theory, which naturally leads to the
question of how much one can say if the transformations in question need
not be ergodic. We have essentially reversed a result of V'yugin and shown
that if an element of the Cantor space is not Martin-Löf random, then
there is a computable function and a computable transformation for which
this element is not typical with respect to the ergodic theorem. More
recently, we have shown that every weakly 2-random element of the Cantor
space is typical with respect to the ergodic theorem for every lower
semicomputable function and computable transformation. I will explain
the proof of the latter result and discuss the technical difficulties
present in producing a full characterization.

#### Presentation

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