Martin Boundaries of Denjoy Domains

Abstract

Let $E( \subset {\mathbf {\hat C}})$ be a compact set in the real axis. It is shown that there exists an $E$ with zero linear measure such that Martin compactification of the domain ${\mathbf {\hat C}} - E$ is not homeomorphic to ${\mathbf {\hat C}}$. Moreover, it is shown that if for some $\lambda > \tfrac {1}{2}$ \[ \frac {{|{E^c} \cap [ - t,t]|}}{t} = O\left ( {{{\left ( {\frac {1}{{\log {t^{ - 1}}}}} \right )}^\lambda }} \right )\quad (t \to 0),\] the set of minimal Martin boundary points of ${\mathbf {\hat C}} - E$ ’over 0’ consists of two points. This assertion is not valid for $\lambda = \tfrac {1}{2}$.