an:00706704
Zbl 0810.30034
Aebischer, Beat; Hong, Sungbok; McCullough, Darryl
Recurrent geodesics and controlled concentration points
EN
Duke Math. J. 75, No. 3, 759-774 (1994).
00022224
1994
j
30F40
M??bius group; limit point; discrete group; recurrent geodesics; Schottky group
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 Poincar?? 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)).
B.Aebischer