Dynamical Borel-Cantelli lemmas.

Anthony Chiu

Frank Adams 1,

The second classical Borel-Cantelli lemma states that, if a
sequence of events are mutually independent and the sum of their
probabilities diverges, then infinitely many of these events occur
almost surely. In the context of dynamical systems, this is not very
interesting because dynamicists rarely work with independent sets.

In this talk, I will explain what it means for a sequence of
measurable sets to be a strongly Borel-Cantelli (SBC) sequence and
what independence should mean when we introduce some dynamics. We will
see what SBC sequences can be in terms of two-sided shift maps,
piecewise expanding maps and, if time permits, maps with indifferent
fixed points.
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