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

We give lower bounds for the size of linearization discs for power series over $\mathbb{C}_p$. For quadratic maps, and certain power series containing a `sufficiently large' quadratic term, we find the exact linearization disc. For finite extensions of $\mathbb{Q}_p$, 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 $\mathbb{C}_p$ on which $f$ is one-to-one). In particular, in unramified extensions of $\mathbb{Q}_p$, 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 \cite{Lindahl:2004,Lindahl:2009,Lindahl:2009eq}. Moreover, it is shown that, for any complete non-Archimedean field, transitivity is preserved under analytic conjugation. Using results by Oxtoby \cite{Oxtoby:1952}, 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 $\mathbb{Q}_p$ 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 $\mathbb{Q}_p$, as well as in any other non-Archimedean field $K$ that is not isomorphic to $\mathbb{Q}_p$ 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.