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

TL;DR

This paper establishes bounds and exact sizes for linearization discs in $p$-adic dynamics, explores conditions for maximal discs, and links these to ergodic and minimal behavior of power series over various non-Archimedean fields.

Contribution

It provides new bounds and exact calculations for linearization discs in $p$-adic dynamics, and connects these to ergodic properties and conjugation invariance.

Findings

Exact linearization discs for quadratic maps.

Maximal linearization discs under certain conditions.

Equivalence of transitivity and unique ergodicity on linearization discs.

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.