Optimal cycles in ultrametric dynamics and minimally ramified power series

We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are optimal in the following sense: They minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. In those cases we exhibit a higher degree polynomial such that all of its cycles are optimal. The proof of these results reveals a connection between the geometric location of periodic points of ultrametric power series and the lower ramification numbers of wildly ramified field automorphisms. We also give an extension of Sens theorem on wildly ramified field automorphisms, and a characterization of minimally ramified power series in terms of the iterative residue.

The size of quadratic $p$-adic linearization disks

We find the exact radius of linearization disks at indifferent fixed points of quadratic maps in $mathbb{C}_p$. We also show that the radius is invariant under power series perturbations. Localizing all periodic orbits of these quadratic-like maps we then show that periodic points are not the only obstruction for linearization. In so doing, we provide the first known examples in the dynamics of polynomials over $mathbb{C}_p$ where the boundary of the linearization disk does not contain any periodic point.

Divergence and convergence of conjugacies in non-Archimedean dynamics

We continue the study in [21] of the linearizability near an indif- ferent fixed point of a power series f, defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz [13] in 1981 that Siegels linearization theorem [27] is true also for non- Archimedean fields. However, they also showed that the condition in Siegels theorem is usually not satisfied over fields of prime character- istic. Indeed, as proven in [21], there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analyt- ically linearizable. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges.

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.

On ergodic behavior of $p$-adic dynamical systems

Monomial mappings, $xmapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The process is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1.