Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits. (English) Zbl 0701.58021

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.


37E99 Low-dimensional dynamical systems
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
28D05 Measure-preserving transformations
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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