Rims of cantor trees
type: Item
instance of: scholarly article
title: Rims of cantor trees
zbMATH Open document ID: 0951.30037
author: J. L. Fernández
author: M. V. Melián
published in: Illinois Journal of Mathematics
publication date: 2000-12-26T00:00:00Z
review text: An abstract graph is a planar tree if it is connected, has no cycles and any pair of edges has no intersection outside the vertices. The authors use along the paper the word tree to denote those planar trees \(T\) whose set \(V\) of vertices is a subset of the open unit disc \({\mathbf D}=\{z\in \mathbb{C}: |z|<1\}\) and whose edges are geodesic arcs in \({\mathbf D}\). If a vertex \(v_0\in V\) is fixed, the elements in \(V\) can be classified according to the graph-distance to \(v_0\), i.e., by defining \(V_0=\{v_0\}\) and for each positive integer \(n\), \[ V_n=\{v\in V:\text{ graph-distance }(v,v_0)=n\}. \] The subset \(A_n\) of the set \(A\) of edges of \(T\) consists on those edges connecting a vertex in \(V_n\) with one in \(V_{n+1}\). The parent \(P(v)\) of a vertex \(v\in V_n\) is the only neighbor of \(v\) in \(V_{n-1}\). The set \[ H(v)=\{w\in V_{n+1}:w\text{ is a neighbor of }v\}, \] is called the set of children of \(v\). For all positive integer \(n\) the following notations are also introduced: \(L_n(T)=\inf \{\text{length} [P(v)v]:v\in V_{n+1}\}\); \(\theta_n (T)=\inf\{ |\text{ang}([vu_1], [vu_2])|\), \(v\in V_n\), \(u_1,u_2\in H(v)\}\), unless \(H(v)\) has one element for all \(v\in V_n\); in such a case \(\theta_n(T)=2\pi\); \(b_n(T)= \sup\{|\text{ang}([P(v)v]\), \([vu])|, v\in V_n\), \(u\in H(v)\}\); \(N_n(T)= \inf\{ \text{card} H(v):v\in V_n\}\). The results in the article refer to trees satisfying the following restrictions: \[ \beta(T)=\sup\{b_n(T):1\leq n\}<\pi/8; \quad \Lambda(T)= \inf\{L_n(T): 1\leq n\}\geq\Lambda_0, \] where \(\Lambda_0>1\) is a certain constant introduced in the paper. An infinite branch emanating from \(v_n \in V_n\) is an infinite sequence of vertices \(\{v_i:n\leq i\}\) such that each \(v_{i+1}\in H(v_i)\). In such case the sequence \(\{v_i:n\leq i\}\) converges in the euclidean topology to a point in the boundary \(\partial\mathbb{D}= \{z\in\mathbb{C}: |z|=1\}\) of \(\mathbb{D}\), which is called the end point of the se…
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 1448995
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: rims
zbMATH Keywords: Cantor tree
zbMATH Keywords: geodesics
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1448995