Linearisation of smooth interval exchange transformations
Ergodic Theory and Dynamical Systems Seminar
31st October 2019, 2:00 pm – 3:00 pm
Fry Building, 2.04
It is probably fair to say that the following result is a basic one taught in most Dynamical Systems courses:
A smooth circle diffeomorphism of irrational rotation number is topologically conjugate to the associated rotation.
A celebrated (and much harder) result of Herman and Yoccoz asserts that as soon as the rotation number satisfies a mild arithmetic condition, this topological conjugacy can be upgraded to a smooth conjugacy.
It is natural to wonder the extent to which the theory of circle diffeomorphisms extends to smooth generalised interval exchange transformations (GIET). In particular, is a smooth GIET of irrational "rotation number" always smoothly conjugate to its linear model?
In this talk I will briefly survey the standard theory of circle diffeomorphisms, try to motivate the relevance of an extension to the case of non-linear interval exchange maps and highlight the main difficulties. If time permits, I will allow myself to advertise my own results on the subject.