Non linearizable holomorphic dynamics having an uncountable number of symmetries

Abstract

We construct germs of holomorphic diffeomorphisms of (C, 0),f(z) =e 2πiα z +O(z 2), α∈R−Q, non linearizable and having an uncountable centralizer, that is an uncountable number of symmetries. This gives a positive answer to a question of M.R. Herman. We construct such examples having no periodic orbit (except 0), and also with a sequence of periodic orbits tending to 0 (distinct from 0). Moreover the examples constructed have an infinite number of finite order symmetries, and the dynamics is quite explicit. We also construct non linearizable analytic diffeomorphisms of the circle having an irrational rotation number and uncountable centralizer with similar properties.