Nonrectifiable level sets for universal covering maps (Q762645)

Let \(\Delta\) denote the open unit disk and let K be a relatively closed subset of \(\Delta\) such that \(O\not\in K\) and \(\Delta\) \(\setminus K\) is connected. Let \(\phi\) denote the universal covering map of \(\Delta\) onto \(\Delta\) \(\setminus K\) with \(\phi (0)=0\). Let \(\gamma\) denote an arbitrary compact rectifiable Jordan arc in \(\Delta\) \(\setminus \{0\}\) and let \(\ell (\cdot)\) denote linear Lebesgue measure. The equivalence \(''\ell (\phi^{-1}(\gamma))=\infty\) if and only if \(\gamma\) meets K'' is valid when K is of capacity zero, but may fail for general K. This paper gives a sufficient (but not necessary) condition for the nonrectifiability of \(\phi^{-1}(\gamma)\). Theorem. If \(\gamma\) contains an irregular boundary point of \(\Delta\) \(\setminus K\), then \(\ell (\phi^{-1}(\gamma))=\infty\). The proof makes essential use of the fact that the Green function of \(\Delta\) \(\setminus K\) has a positive fine limit at an irregular boundary point.