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

Lalley, Steven P.

Acta Math. 163, No. 1-2, 1-55 (1989).

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.

MSC:

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)

Keywords:

shifts; suspension flows; thermodynamic; periodic orbits; perturbation theory; Schottky groups; Fuchsian groups; geodesic flow; noneuclidean lattice points; limit set; random walk; Hausdorff measure; renewal theorem; symbolic dynamics

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