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The action at infinity of conservative groups of hyperbolic motions need not have atoms

Published online by Cambridge University Press:  19 September 2008

John A. Velling  and
Katsuhiko Matsuzaki
John A. Velling
Affiliation:
Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, New York 11210, USA
Katsuhiko Matsuzaki
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan
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Abstract

Herein the authors show that discrete groups of motions on Hn+1 may be conservative on Sn but have no positive measure ergodic components for this boundary action. An explicit example of such a group is given for H3 using the Apollonian circle packing of R2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 3 , September 1991 , pp. 577 - 582
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

[1]Agard, S.. Mostow rigidity on the line: A survey. Holomorphic Functions and Moduli II, MSRI 11 (1988), 1–12.CrossRefGoogle Scholar
[2]Hopf, E.. Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull. Amer. Math. Soc. 77 (1971), 863–887.Google Scholar
[3]Lyons, T. & Sullivan, D.. Function theory, random paths and covering spaces. J. Diff. Geom. 19 (1984), 299–323.Google Scholar
[4]Millson, J.. On the first Betti number of a constant negatively curved manifold. Ann. Math. 104 (1976), 235–247.Google Scholar
[5]Pommerenke, C.. On the Green's function of Fuchsian groups. Ann. Acad. Sci. Fenn. ser. A. I. 2 (1976), 409–427.CrossRefGoogle Scholar
[6]Pommerenke, C.. On Fuchsian groups of accessible type. Ann. Acad. Sci. Fenn. ser. A. I. 7 (1982), 249–258.CrossRefGoogle Scholar
[7]Sullivan, D.. On the ergodic theory at infinity of an arbitrary group of hyperbolic motions. Riemann Surfaces and Related Topics: Proc. 1978 Stony Brook Conference. Ann. Math. Stud. 97 (1981), 465–496.CrossRefGoogle Scholar
[8]Taniguchi, M.. Examples of discrete groups of hyperbolic motions conservative but not ergodic at infinity. Ergod. Th. & Dynam. Sys. 8 (1988), 633–636.Google Scholar
[9]Velling, J.. Recurrent Fuchsian groups whose Riemann surfaces have infinite dimensional spaces of bounded harmonic functions. Proc. Jap. Acad. 65 ser.A (1989), 211–214.Google Scholar
[10]Veiling, J.. Green's functions and normal covers of hyperbolic manifolds. In preparation.Google Scholar