Rims of cantor trees
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 sequence \(\{v_i:n\leq i\}\). The rim Rim\((T)\) of \(T\) is the set of all endpoints of all infinite branches of the three \(T\). It is a compact subset of \(\partial{\mathbf D}\) and for \(\eta\in[0,1]\) its \(\eta\)-content is defined by \[ \text{Cont}_\eta \bigl(\text{Rim} (T)\bigr)= \sum_i\text{infr}^\eta_i, \] where the infimum is taken over all coverings of \(\text{Rim} (T)\) by balls of radius \(r_i\) in \(\partial {\mathbf D}\). Recall that the Hausdorff dimension \(\dim\text{(Rim}(T))\) is defined as \[ \text{dim} \bigl(\text{Rim}(T)\bigr)= \sup\biggl\{\eta: \text{Cont}_\eta \bigl(\text{Rim}\bigl(T) bigr)> 0\biggr\}. \] The authors also introduce the notion of Cantor tree, which is rather technical, and the order \((T)\) of the Cantor tree \(T\). The main result of the paper is the following: Theorem: If \(T\) is a Cantor tree of order \(\delta\), then \(\text{Rim} (T)\) has non zero \(\eta\)-content for all \(\eta<\delta\). In particular it follows that \[ \text{order}(T)\leq\dim \bigl(\text{Rim}(T)\bigr). \] The authors announce an application of this theorem to estimate the size of the set of escaping geodesics in a Riemann surface to appear in a subsequent paper. The proofs are very technical but clearly written, and also the structure of the article is very transparent.
scientific article; zbMATH DE number 1448995
scientific article