Dynamics of Multidimensional Continued Fraction Algorithms as a Random Walk on Graphs

Motivated by the richness of the Gauss algorithm, which allows for the efficient calculation of the best approximations of a real number by rationals, many mathematicians have proposed generalizations of these algorithms to approximate vectors of dimension greater than one. Examples include the algorithm
introduced by Poincaré at the end of the 19th century and those by Brun and Selmer in the mid-20th century.

From the early 1990s to the present, there has been significant research studying the convergence of these algorithms. Notably, Schweiger and Broise have demonstrated that the Selmer and Brun algorithms are convergent and ergodic. Perhaps more surprisingly, Nogueira has shown that the algorithm proposed by
Poincaré almost never converges. More recently these renormalisations process have been used to study families of fractals sets appearing in number theory or the so called Rauzy gasket.

Starting from the classic case of the Farey algorithm, which is an "additive" version of the Gauss algorithm, I will present a combinatorial perspective on these algorithms that allows for a transition from a deterministic view to a probabilistic approach. In this model, selecting a random vector according to the Lebesgue measure corresponds to following a random walk with memory on a labeled graph called a simplicial system. The laws governing this random walk are elementary, and we can develop probabilistic techniques to study their generic dynamic behavior. This will lead us to describe a purely graph-theoretic criterion to demonstrate whether a continued fraction algorithm converges or not.