Harmonic dimensions of covering surfaces (Q1339807)

Let \(F\) be an open Riemann surface of null boundary which has a single ideal boundary component in the sense of Kerékjártó-Stoïlow. A relatively noncompact subregion \(\Omega\) of \(F\) is said to be an end of \(F\) if the relative boundary \(\partial \Omega\) consists of finitely many analytic Jordan curves. We denote by \({\mathcal P} (\Omega)\) the class of all nonnegative harmonic functions on \(\Omega\) with vanishing values on \(\partial \Omega\). The harmonic dimension of \(\Omega\), \(\dim {\mathcal P} (\Omega)\) in notation, is defined as the minimum number of elements of \({\mathcal P} (\Omega)\) generating \({\mathcal P} (\Omega)\) provided that such a finite set exists, otherwise as \(\infty\). It is well-known that \(\dim {\mathcal P} (\Omega)\) does not depend on a choice of end of \(F:\dim {\mathcal P} (\Omega) = \dim {\mathcal P} (\Omega')\) for any pair \((\Omega, \Omega')\) of ends of \(F\). In terms of the Martin compactification \(\dim {\mathcal P} (\Omega)\) coincides with the number of minimal points over the ideal boundary. In this note we especially concern with ends \(W\) which are subregions of \(p\)-sheeted unlimited covering surfaces of \(\{0 < | z | \leq \infty\}\). For these \(W\) it is known that \(1 \leq \dim {\mathcal P} (W) \leq p\). Consider two positive sequences \(\{a_ n\}\) and \(\{b_ n\}\) satisfying \(b_{n+1} < a_ n < b_ n < 1\) and \(\lim_{n \to \infty} a_ n = 0\). Set \(G = \{0 < | z | < 1\} - I\) where \(I = \cup^ \infty_{n = 1} I_ n\) and \(I_ n = [a_ n, b_ n]\). We take \(p(>1)\) copies \(G_ 1, \dots, G_ p\) of \(G\). Joining the upper edge of \(I_ n\) on \(G_ j\) and the lower edge of \(I_ n\) on \(G_{j+1} (j \bmod p)\) for every \(n\), we obtain a \(p\)-sheeted covering surface \(W=W^ I_ p\) of \(\{0 < | z | < 1\}\) which is naturally considered as an end of a \(p\)-sheeted covering surface of \(\{0 < | z | \leq \infty\}\). Then we proved the following. Theorem. Suppose that \(p = 2^ m (m \in \mathbb{N})\). Then (i) \(\dim {\mathcal P} (W) = p\) if and only if \(I\) is thin at \(z = 0\); (ii) \(\dim {\mathcal P} (W) = 1\) if and only if \(I\) is not thin at \(z=0\).