Recurrent Fuchsian groups whose Riemann surfaces have infinite dimensional spaces of bounded harmonic functions (Q921172)

A Fuchsian group \(\Gamma\) acting on the unit disc D is called recurrent if for any positive measure subset \(A\subset S^ 1\) there exist infinitely many \(\gamma\in \Gamma\) such that \(A\cap \gamma A\) has positive Lebesgue measure. The paper under review contributes to the function theory corresponding to such groups, especially to the structure of the spaces of bounded harmonic functions on \({\mathcal R}=D/\Gamma\). Motivated by interesting examples, Taniguchi conjectured: If \(\Gamma\) is recurrent then \({\mathcal R}=D/\Gamma\) is in \(O^{\infty}_{HB}\). The author gives examples showing that this conjecture is false. - The surface \({\mathcal R}=D/\Gamma\) is said to be in \(O^ n_{HB}\) if up to sets of measure zero the action of \(\Gamma\) decomposes \(S^ 1\), into at most n disjoint positive measure ergodic components. The author constructs a surface \({\mathcal R}\in O^{\infty}_{HB}\setminus \cup^{\infty}_{n=1}O^ n_{HB}\) (see Theorem 2).