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Recurrent geodesics and controlled concentration points. (English) Zbl 0810.30034
A complete hyperbolic manifold is a quotient $$M= B/\Gamma$$, where $$B\subset\mathbb{R}^ m$$ is the open unit ball with the Riemannian metric $$ds= {2| dx|\over 1-| x|^ 2}$$, i.e. the Poincaré model of hyperbolic space, and $$\Gamma$$ is a discrete torsion free group of hyperbolic isometries. An oriented geodesic $$\beta: \mathbb{R}\to M$$ parametrized by arclength is recurrent if for all (equivalently for some) $$t_ 0\in \mathbb{R}$$ there is a sequence $$t_ n\to \infty$$ such that $$\beta'(t_ n)$$ converges to $$\beta'(t_ 0)$$ in the unit tangent bundle $$T_ 1 M$$. It is shown that a geodesic $$\beta$$ is recurrent if and only if it is approximable by closed geodesics, i.e. for all $$t_ 0\in \mathbb{R}$$ and $$\varepsilon> 0$$ there exists a closed geodesic $$\gamma: \mathbb{R}\to M$$ such that $$\rho(\gamma(t),\beta(t_ 0+ t))\leq \varepsilon$$ for $$0\leq t\leq\text{length}(\gamma)$$. (Obviously, $$\text{length}(\gamma)\to \infty$$ when $$\varepsilon\to 0$$, unless $$\beta$$ is closed.) As a corollary, the recurrent geodesics form a $$G_ \delta$$-subset $$\mathcal R$$ of the space of all geodesics (which can be identified with $$T_ 1 M$$), hence $$\mathcal R$$ is topologically complete.
The endpoints of lifts to $$B$$ of recurrent geodesics in $$M$$ are characterized as the special limit points of $$\Gamma$$ mentioned in the title. A limit point of $$\Gamma$$ is a controlled concentration point $$(p\in \Lambda_{ccp})$$ if there is a neighborhood $$U\subset \partial B$$ of $$p$$ such that for every neighborhood $$V$$ of $$p$$ there exists $$g\in \Gamma$$ satisfying $$g(U)\subseteq V$$ and $$p\in g(V)$$. The paper gives several other characterizations of controlled concentration points similar to characterizations of conical limit points. For instance, $$p\in \Lambda_{ccp}$$ if and only if $$\exists(g_ n)\subset\Gamma: g_ n(p)\to p$$ and $$g_ n(0)\to r\in \partial B\backslash\{p\}$$. Clearly, $$\Lambda_{ccp}$$ is a subset of the set of conical limit points $$\Lambda_ c$$, and it follows from known results that $$\Lambda_{ccp}$$ has full or zero measure according to whether $$\Gamma$$ is of divergence or of convergence type. A comparison of the above definition of $$\Lambda_{ccp}$$ with the following ‘concentration characterization’ of $$\Lambda_ c$$ (which the third author found after the paper was published) nicely shows the difference between these two kinds of limit points: $$p\in \Lambda_ c$$ if and only if there exist open neighborhoods $$U_ 1$$ and $$U_ 2$$ of $$p$$ such that $$\overline U_ 2\subset U_ 1$$ and for every neighborhood $$V$$ of $$p$$ there exists $$g\in \Gamma$$ with $$g(U_ 1)\subseteq V$$ and $$p\in g(U_ 2)$$.
The last section contains a Schottky group example showing that $$\Lambda_ c\backslash\Lambda_{ccp}$$ can be uncountable. (To correct a misprint in Proposition 1.8, replace ‘distance $$\beta$$’ by ‘distance $$\delta$$’ just before (i)).
Reviewer: B.Aebischer

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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##### References:
  B. Aebischer, The limiting behavior of sequences of Möbius transformations , Math. Z. 205 (1990), no. 1, 49-59. · Zbl 0686.40002 · doi:10.1007/BF02571224 · eudml:174158  B. Aebischer, The limiting behavior of sequences of quasiconformal mappings , Canad. Math. Bull. 33 (1990), no. 4, 494-502. · Zbl 0699.30016 · doi:10.4153/CMB-1990-079-0  S. Agard, A geometric proof of Mostow’s rigidity theorem for groups of divergence type , Acta Math. 151 (1983), no. 3-4, 231-252. · Zbl 0532.30038 · doi:10.1007/BF02393208  L. V. Ahlfors, Möbius transformations in several dimensions , Ordway Professorship Lectures in Math., University of Minnesota School of Mathematics, Minneapolis, Minn., 1981. · Zbl 0517.30001  A. Beardon, The geometry of discrete groups , Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. · Zbl 0528.30001  A. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra , Acta Math. 132 (1974), 1-12. · Zbl 0277.30017 · doi:10.1007/BF02392106  A. Casson and S. Bleiler, Automorphisms of surfaces after Nielsen and Thurston , London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. · Zbl 0649.57008  J. Dugundji, Topology , Allyn and Bacon Inc., Boston, Mass., 1966. · Zbl 0144.21501  J. B. Garnett, F. W. Gehring, and P. W. Jones, Quasiconformal groups and the conical limit set , Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) ed. E. Drasin, et al., Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 59-67. · Zbl 0654.30013  F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I , Proc. London Math. Soc. (3) 55 (1987), no. 2, 331-358. · Zbl 0628.30027 · doi:10.1093/plms/s3-55_2.331  S. Hong, Controlled concentration points and groups of divergence type , to appear in proceedings of Low-dimensional Topology Conference, Knoxville, Tennessee, 1992. · Zbl 0872.22007  B. Maskit, Kleinian Groups , Grundlehren Math. Wiss., vol. 287, Springer-Verlag, Berlin, 1987.  D. McCullough, Weak concentration points for discrete groups of Möbius transformations , to appear in Illinois J. Math. · Zbl 0805.20041  D. McCullough, Concentration points for Fuchsian groups , · Zbl 0959.20046 · doi:10.1016/S0166-8641(99)90064-0  P. Nicholls, Transitive horocycles for Fuchsian groups , Duke Math. J. 42 (1975), 307-312. · Zbl 0348.30018 · doi:10.1215/S0012-7094-75-04228-3  P. Nicholls, The Ergodic Theory of Discrete Groups , London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. · Zbl 0674.58001
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