Quasiconformal mappings and minimal Martin boundary of $p$-sheeted unlimited covering surfaces of the once punctered Riemann sphere $\hat{\mathbb{C}} \setminus \{0\}$ of Heins type

Abstract

Let $R$ and $R'$ be $p$-sheeted unlimited covering surfaces of the once punctured Riemann sphere $\hat{\mathbb{C}} \setminus \{0\}$ of Heins type which are quasiconformal equivalent to each other. Then the cardinal numbers of minimal Martin boundaries of $R$ and $R'$ are same.

Let $R$ be a 2-sheeted unlimited covering surface of the once punctured Riemann sphere $\hat{\mathbb{C}} \setminus \{0\}$ of Heins type and $R'$ be an open Riemann surface. If $R$ and $R'$ are quasiconformal equivalent to each other and the set of branch points of $R$ satisfies a condition, then the cardinal numbers of minimal Martin boundaries of $R$ and $R'$ are same.