Let N be an integer greater than 1. We will study the set of
N-tuples (A1, A2,.., AN) of matrices in SL(2,R) such that the
norm of words in A1, A2,.. AN (with no inverses allowed) grow
exponentially with the length. This open set is well-understood
when N=2; for instance, in this case, the set of connected components
is naturally in 1-to-1 correspondence with rational numbers
in (0,1). When N is at least 3, new phenomena occur , leading
to several open questions. This is jointwork with Artur Avila
and Jairo Bochi.
Jean-Christophe Yoccoz received the Agrégation de
Mathématiques in 1977 in joint first position. He was
a student of M. Herman and submitted his doctoral thesis in
1985. As a student of Herman, a leading word expert on dynamical
systems, it was not surprising that Yoccoz would himself work
on dynamical systems. In his thesis, Yoccoz improved theorems
of his supervisor Herman by giving simpler proofs but also
obtaining the same results under weaker hypotheses.
He was appointed as professor at the University of Paris-Sud
(Orsay). He became a member of the Institut Universitaire
de France and a member of the Unité Recherche Associé
"Topology and Dynamics" of the Centre National de
la Recherche Scientifique at Orsay.
At the International Congress of Mathematicians in Zurich
in 1994, Yoccoz received his greatest honour for this work
on dynamical systems when he was awarded a Fields Medal.