Arithmetic properties of centralizers of diffeomorphisms of the half-line

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r_f its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1_f is always a one-parameter group, naturally identified to \R, (with f identified to 1). On the other hand, Z^r_f, for r greater or equal to 2, can be smaller: in [Se], Sergeraert constructed an f whose C^infty centralizer reduces to the infinite cyclic group generated by f (i.e Z^\infty_f identifies to \Z). In [Ey1], we adapted Sergeraert's construction to obtain an f whose C^r centralizer, for all r between 2 and \infty, contains a Cantor set K but is still strictly smaller than Z^1_f (= \R). Here, we improve [Ey1] to construct, for any Liouville number alpha, an f as above such that, in addition, alpha belongs to K.