Dynamical analysis of particular trajectories in the Euclid system
Salle de séminaire M.0.1
(Universidad Nacional de Gral, Sarmiento and CONICET, Argentina)
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.