Small denominators in complex p-adic dynamics

Abstract

We study the problem of small denominators in the field of complex p-adic numbers C p . We prove that, in fact, we can always obtain the estimate from below D n ( x) ≥ CA n , C( x), A( x) > 0, for the denominator D n ( x) which is used in the construction of a congugate map for a dynamical system f having the derivative x = f′( a) in the fixed point a. Moreover, we find (via a long chain of technical computations) forms of coefficients C( x) and A( x). These are complicated functions of x which depend on the relative position of x and p sth roots of unity. Our result can be used to find radius of convergence for conjugate maps for C p -analytic dynamical systems at neutral fixed points (or cycles).