Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

In this talk, we  study  the probabilistic behavior of particular trajectories (finite or periodic) of a  given dynamical system.
For these particular trajectories, ergodic theorems  do  not apply, 
and we  explain the main principles  of  the Dynamical Analysis Method, in the case of the Gauss map.
In this case, finite trajectories  coincide with rational trajectories, and  thus executions of the Euclid algorithm.  
Then, the analysis of the  probabilistic  performance  of  the Euclid algorithm  is strongly related to  the  probabilistic study  of rational trajectories.
The Dynamical Analysis Method mixes general principles of Analytic Combinatorics (generating functions, Tauberian theorems)
with techniques and tools  that come from Dynamical Systems Theory (transfer operator). 
The transfer operator is viewed  here as  a generating operator, that  generates itself generating functions,
that are  in this case of Dirichlet type. 
Finally, with generating functions at hand, together with dominant spectral properties of the transfer operator, 
it is possible to extract  (via  Tauberian theorems), precise information  about the behaviour of finite (and periodic) trajectories.