##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Yves Lacroix

#### Universite du Sud Toulon-Var

## On the law of series in ergodic theory

##### Abstract:

In a discrete time ergodic aperiodic dynamical system, one can look at the recurrence times for small measure sets. As the measure of these sets decrease to zero, within several regular enough setups, one has proved that the normalized hitting time distributions or return time distributions tend to the exponential law. Such limiting distribution is called an asymptotic distribution. We have proved with T. Downarowicz that in a positive entropy system any asymptotic along cylinder sets (whence for symbolic dynamical systems) must be subexponential. This turns out to be interpreted as a rule saying that clusters (grouped repetitions of visits to a set) are whatever at least as large as they can be in the independent process case, where their appearances are ruled by the exponential distribution. We also present some material saying that typically a symbolic dynamical system should present along a sequence of lengths of cylinders of upper density one, absolute clustering.

Host: Jason Schweinsberg

### October 16, 2008

### 10:00 AM

### AP&M 6402

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