Positive Harmonic Functions on Denjoy Domains in the Complex Plane

Let $\Om$ be a domain in the complex plane $\C$ whose complement $E=\OC\setminus \Om$, where $\OC=\C\cup\{\infty\}$ is a subset of the real line (i.e. $\Om$ is a Denjoy domain). If each point of $E$ is regular for the Dirichlet problem in $\Om$, we provide a geometric description of the structure of $E$ near infinity such that the Martin boundary of $\Om$ has one or two "infinite" points.