Mathematics

Traditionally in ergodic theory one deals with a measure-preserving map of a space of total measure one.
This allows for a probability interpretation, withsubsets being "events" of probability equal to their measure.
The Birkhoff Ergodic Theorem says that if the map is measure-theoretically indecomposable
("ergodic")then the subset of the integers which records the times of return of almost any point to
the set has a Cesaro density equal to its measure: that"time average equals space average".
But sets of zero density can occur naturally in ergodic theory and probability as well:
indeed given a recurrent ergodic map of an infinite measure space, the times of return to a set of finite measure must have density
zero. Nevertheless for nice cases, this set can exhibit additional geometric structure: a "Hausdorff dimension" between 0 and 1, and a Hausdorff measure
of that dimension. This last leads to a new statement of the type "time average equals space average": an order-two ergodic theorem.

We will survey some examples coming from probability theory (renewal processes) and from dynamics (adic transformations).
Interesting example of this last come from return maps to measure-zero Cantor subsets of a circle rotation.

Joint work with Marina Talet.