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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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