Application of Poincaré recurrences to the study of mixed dynamical systems

The Poincaré recurrence theorem played a crucial role in the
development of statistical mechanics at the end of the 19th century.
In more recent years, Poincaré recurrences received a growing
attention as a subject of the theory of dynamical systems.

In this respect, I will present a study of the statistics of first return
times for a skew integrable map and for mixed dynamical systems.
I will consider, in particular, a model obtained by coupling a regular
and a mixing map together, showing that it allows to understand
the behaviour of Poincaré recurrences for a system of more physical
interest, the so called \"standard map\".

I will also describe an extension of the notion of statistics of first
return times, namely the distributions of the number of visits, whose
definition is based on successive return times. Finally, I will present
an application of Poincaré recurrences to the analysis of the coding
and noncoding regions of genomic sequences.