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 ${\mathbb{C}}_p$. 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.

Revised version of preprint 04098 MSI, 2004, Växjö University, Sweden and part of the authors PhD thesis "On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point", Växjö University, 2007

References in corpus (3)

• Linear Fractional p-Adic and Adelic Dynamical Systems
• $p$-adic discrete dynamical systems and their applications in physics and cognitive sciences
• Analysis and dynamics on the Berkovich projective line