On almost reducibility in quasi-periodic dynamics (Raphaël Krikorian, Paris)

13.10.2016 16:15

An important question in dynamics is whether a given dynamical system can be conjugated to a simpler one (a model dynamics). Though in general the conjugacy classes of a given dynamical systems are too numerous, some classes of quasi-periodic systems (such as for example circle diffeomorphisms) can be conjugated to simple models (such as rotations in the case of circle diffeomorphisms). A fundamental theorem in that direction is the Herman-Yoccoz theorem that claims that any smooth orientation preserving orientation preserving diffeomorphism with a diophantine rotation number can be conjugated to a rigid rotation. I will address in this talk a weaker notion, that of almost-reducibility. In the case of circle diffeomorphisms it is defined as follows: a smooth diffeomorphism of the circle $f$ with rotation number $\alpha$ is almost reducible if there exists a sequence of smooth conjugations $g_n$ such that $g_n^{-1}\circ f\circ g_n$ converges in the smooth topology to $x\mapsto x+\alpha$. I will discuss what can be said about almost reducibility of smooth circle diffeomorphisms and I will also discuss similar problems for pseudo-rotations of the disk (orientation and area preserving diffeomorphisms of the 2-disk fixing the origin and the boundary without fixed points).


P.S. The colloquium will be followed by an aperitif

Lieu

Room 17, Acacias, Colloque

entrée libre