Estimates of linearization discs in $p$-adic dynamics with application to ergodicity

Abstract

We give lower bounds for the size of linearization discs for power series over Cp. For quadratic maps, and certain power series containing a 'sufficiently large' quadratic term, we find the exact linearization disc. For finite extensions of Qp, we give a sufficient condition on the multiplier under which the corresponding linearization disc is maximal (i.e. its radius coincides with that of the maximal disc in Cp on which f is one-to-one). In particular, in unramified extensions of Qp, the linearization disc is maximal if the multiplier map has a maximal cycle on the unit sphere. Estimates of linearization discs in the remaining types of non-Archimedean fields of dimension one were obtained in [44,46,47].Moreover, it is shown that, for any complete non-Archimedean field, transitivity is preserved under analytic conjugation. Using results by Oxtoby [52], we prove that transitivity, and hence minimality, is equivalent the unique ergodicity on compact subsets of a linearization disc. In particular, a power series f over Qp is minimal, hence uniquely ergodic, on all spheres inside a linearization disc about a fixed point if and only if the multiplier is maximal. We also note that in finite extensions of Qp, as well as in any other non-Archimedean field K that is not isomorphic to Qp for some prime p, a power series cannot be ergodic on an entire sphere, that is contained in a linearization disc, and centered about the corresponding fixed point.