Wandering domains and nontrivial reduction in non-archimedean dynamics

Wandering domains in non-archimedean polynomial dynamics

We extend a recent result on the existence of wandering domains of polynomial functions defined over the p-adic field C_p to any algebraically closed complete non-archimedean field C_K with residue characteristic p>0. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree p+1 polynomials in C_K[z].

Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

Let $f$ be a rational function of degree $d>1$ on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of $f$, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for $C^1$-test functions in terms of the proximity of wandering critical orbits to the initial points, an…

非阿基米德动力学中的无理 Fatou 分量

Irrational Fatou components in non-Archimedean dynamics

本文通过显式计算游荡域，首次给出了非阿基米德动力学中无理盘类型的 Fatou 分量的例子。

Non-archimedean Yomdin-Gromov parametrizations and points of bounded height

We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C} ((t))$, in analogy to results by Pila and Wilkie, resp. by Bombieri…

关于赋值域的两点注记

Two notes on valued fields

研究赋值域中由绝对值诱导的度量维数以及乘法的均匀开性，证明了非阿基米德情形下度量维数等于密度特征，阿基米德情形下取决于子域类型，并证明乘法在任意赋值域上均匀开。

Non-invariance of weak approximation with Brauer-Manin obstruction for surfaces

In this paper, we study the property of weak approximation with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension of number fields L/K, assuming a conjecture of M. Stoll, we construct a smooth, projective, and geometrically connected surface over K such that it satisfies weak approximation with Brauer-Manin obstruction off all archimedean places, while its base change to L fails. Then we illustrate this construction with an explicit unconditional example.