Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits
This paper outlines a new approach to the asymptotic analysis of certain counting functions arising in the geometry of discrete groups. The approach is based on an analogue of the renewal theorem for counting measures in symbolic dynamics. The counting problems considered in this paper are mostly tied up with the ergodic behavior of the action of a discrete group at \(\infty\). Some of these problems may be solved by other methods of noncommutative harmonic analysis, e.g., the Selberg trace formula, and in these cases the alternative methods may give sharper results (especially error estimates). Also, the methods developed here are not well suited for groups with parabolic elements, because of difficulties with the symbolic dynamics. However, our approach is suitable for certain problems that are apparently outside the scope of noncommutative harmonic analysis, in particular, problems directly concerned with the geometry of the limit set.
scientific article; zbMATH DE number 4149189
scientific article