Topological characterizations of certain limit points for Möbius groups (Q2770327)

The authors give some characterizations of several kinds of limit points in the case of two generator Schottky groups acting on the Poincaré disc \(B^2\). They take \(\Gamma\) to be a discrete subgroup of the group of hyperbolic isometries acting on the Poincaré disc \(B^m\), \(m\geq 2\). Here, the set \(\Lambda(\Gamma)=\Lambda\) denotes the limit set of \(\Gamma\) and \(\text{CH}(\Lambda)\) denotes the convex hull of \(\Gamma\), where \(\Gamma\) is a non-elementary group.NEWLINENEWLINENEWLINEA limit point \(p\) is called a Myrberg-Agard density point for \(\Gamma\) if whenever \(\mu\) is an oriented geodesic for \(\Gamma\) and \(\alpha\) is a geodesic ray ending at the point \(p\) in \(\text{CH}(\Lambda)\), there is a sequence of elements \(\{\gamma_i\}\) such that \(\{\gamma_i(\alpha)\}\) converges to \(\mu\) in an oriented sense. With each \(\alpha\), a special sequence \(S(\alpha)\) is associated. With this terminology, the following theorem is obtained as the main result of the work:NEWLINENEWLINENEWLINETheorem: A necessary and sufficient condition for a limit point \(p\) to be a Myrberg-Agard density point is that for every ray \(\alpha\) ending at \(p\), every admissible sequence appears as a sequence of \(S(\alpha)\).