A look at nonexpansive mappings in non-Archimedean vector spaces

In a spherically complete ultrametric space every nonexpansive self-mapping T has a fixed point ̄x or a minimal invariant ball B(̄x, d(̄x, T(̄x)). We show how we can approximate this fixed center ̄x in a non-Archimedean vector space. And, we give a synthetic study for increasing mapping in non-Archimedean local fields.

Poisson approximation related to spectra of hierarchical Laplacians

Let [Formula: see text] be a locally compact separable ultrametric space. Given a measure [Formula: see text] on [Formula: see text] and a function [Formula: see text] defined on the set of all non-singleton balls [Formula: see text] of [Formula: see text], we consider the hierarchical Laplacian [Formula: see text]. The operator [Formula: see text] acts in [Formula: see text] is essentially self-adjoint and has a purely point spectrum. Choosing a sequence [Formula: see text] of i.i.d. random variables, we consider the perturbed function [Formula: see text] and the perturbed hierarchical Laplacian [Formula: see text] Under certain conditions, the density of states [Formula: see text] exists and it is a continuous function. We choose a point [Formula: see text] such that [Formula: see text] and build a sequence of point processes defined by the eigenvalues of [Formula: see text] located in the vicinity of [Formula: see text]. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity [Formula: see text].

On the rate of convergence in the central limit theorem for hierarchical Laplacians

Let (X,d) be a proper ultrametric space. Given a measuremonXand a functionB↦C(B) defined on the set of all non-singleton ballsBwe consider the hierarchical LaplacianL=LC. Choosing a sequence {ε(B)} of i.i.d. random variables we define the perturbed functionC(B,ω) and the perturbed hierarchical LaplacianLω=LC(ω). We study the arithmetic means λ̅(ω) of theLω-eigenvalues. Under certain assumptions the normalized arithmetic means (λ̅−Eλ̅) ∕ σ(λ̅) converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.

Fixed point theorem in ultrametric space

We give a class of generalizations of a number of known fixed point theorems to mappings defined on an ultrametric space and non-Archimedean normed space which are endowed with a graph. We also investigate the relationship between weak connectivity and the existence of fixed points for these mappings.

Finiteness Properties of Some Groups of Local Similarities

Hughes has defined a class of groups that we call finite similarity structure (FSS) groups. Each FSS group acts on a compact ultrametric space by local similarities. The best-known example is Thompson’s group V. Guided by previous work on Thompson’s group, we show that many FSS groups are of type F∞. This generalizes work of Ken Brown from the 1980s.

LOCALLY CONSTANT FUNCTIONS IN C-MINIMAL STRUCTURES

Let M be a C-minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → T in C-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a description of definable subsets of T in one variable and analogues of known results in algebraically closed valued fields.

ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

Motivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].