Institute of Mathematics


Modul:   MAT760  Ergodic Theory and Dynamical Systems Seminar

Dynamics of simplicial systems and multi-dimensional continued fraction algorithms

Talk by Dr. Charles Fougeron

Date: 18.01.21  Time: 15.00 - 16.15  Room: Y27H28

Motivated by the richness of the Gauss algorithm which allows to efficiently compute the best approximations of a real number by rationals, many mathematicians have proposed generalisations of these algorithms to approximate vectors of dimension greater than 1. Examples include Poincaré's algorithm introduced at the end of the 19th century or those of Brun and Selmer in the middle of the 20th century. Since the beginning of the 1990s to the present day, there has been a number of works studying the convergence of these algorithms. In particular, Schweiger and Broise have shown that the algorithms of Selmer and Brun are convergent and ergodic. Perhaps more surprisingly, Nogueira demonstrated that the algorithm proposed by Poincaré almost never converged. Starting from the classical case of Farey's algorithm, which is an "additive" version of Gauss's algorithm, I will present a combinatorial point of view on these algorithms which allows the passage from a deterministic view to a probabilistic approach. Indeed, in this model, taking a random vector for the Lebesgue measurement will correspond to following a random walk with memory in a labelled graph called symplicial system. The laws of probability for this random walk are elementary and we can thus develop probabilistic techniques to study their generic dynamical behaviour. This will lead us to describe a purely graph theory criterion to demonstrate the convergence or not of a continuous fraction algorithm.