Velling, John A. Recurrent Fuchsian groups whose Riemann surfaces have infinite dimensional spaces of bounded harmonic functions. (English) Zbl 0709.30038 Proc. Japan Acad., Ser. A 65, No. 7, 211-214 (1989). 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). Reviewer: J.Elstrodt Cited in 1 Document MSC: 30F20 Classification theory of Riemann surfaces 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) Keywords:Fuchsian group PDFBibTeX XMLCite \textit{J. A. Velling}, Proc. Japan Acad., Ser. A 65, No. 7, 211--214 (1989; Zbl 0709.30038) Full Text: DOI References: [1] Agard, S.: Mostow rigidity on the line. A survey. Holomorphic Functions and Moduli II, MSRI 11, 1-12 (1988). · Zbl 0657.30034 [2] Constantinescu, C. and Cornea, A.: Ideale Rander Riemannischer Flaschen. Springer-Verlag, Berlin (1963). [3] Constantinescu, C. and Cornea, A.: Uber den idealen Rand und einige seiner Anwendungen bei der Klassifikation der Riemannschen Fliischen. Nagoya Math. J., 13, 169-233 (1958). · Zbl 0093.07701 [4] Hopf, E.: Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull. Amer. Math. Soc, 77, 863-887 (1971). · Zbl 0227.53003 [5] Lyons, T. and Sullivan, D.: Function theory, random paths and covering spaces. J. Diff. Geom., 19, 299-323 (1984). · Zbl 0554.58022 [6] Mori, A.: A note on unramified abelian covering surfaces of a closed Riemann surface. J. Math. Soc. Japan, 6, 162-176 (1954). · Zbl 0059.07003 [7] Pommerenke, C.: On Fuchsian groups of accessible type. Ann. Acad. Sci. Fenn., ser. A. I. 7, 249-258 (1982). · Zbl 0471.30036 [8] Rubel, L. and Ryff, J.: The bounded weak-star topology and the bounded analytic functions. J. Func. Anal., 5, 167-183 (1970). · Zbl 0189.44701 [9] Sullivan, D.: On the ergodic theory at infinity of an arbitrary group of hyperbolic motions. Riemann surfaces and related topics. Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Studies, 97, 465-496 (1981). · Zbl 0567.58015 [10] Sario, L. and Nakai, M.: Classification theory of Riemann surfaces. Springer-Verlag, Berlin (1970). · Zbl 0199.40603 [11] Taniguchi, M.: Examples of discrete groups of motions conservative but not ergodic at infinity. Ergod. Th. and Dynam. Sys., 8, 633-636 (1988). · Zbl 0643.22006 [12] Tsuji, M.: Potential Theory in Modern Function Theory. Maruzen, Tokyo (1959). · Zbl 0087.28401 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.