Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits
type: Item
instance of: scholarly article
title: Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits
zbMATH Open document ID: 0701.58021
DOI: 10.1007/BF02392732
published in: Acta Mathematica
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 4149189
zbMATH Keywords: shifts
zbMATH Keywords: suspension flows
zbMATH Keywords: thermodynamic
zbMATH Keywords: periodic orbits
zbMATH Keywords: perturbation theory
zbMATH Keywords: Schottky groups
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: geodesic flow
zbMATH Keywords: noneuclidean lattice points
zbMATH Keywords: limit set
zbMATH Keywords: random walk
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: renewal theorem
zbMATH Keywords: symbolic dynamics
author: Steven P. Lalley
MaRDI profile type: Publication
Wikidata QID: Q125862750
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description: scientific article; zbMATH DE number 4149189