# Groups of automorphisms of trees and their limit sets

@article{Hersonsky1997GroupsOA, title={Groups of automorphisms of trees and their limit sets}, author={Sa'ar Hersonsky and John H. Hubbard}, journal={Ergodic Theory and Dynamical Systems}, year={1997}, volume={17}, pages={869-884} }

Let T be a locally finite simplicial tree and let ! ⊂ Aut(T ) be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincare series associated with !, which is also the Hausdorff dimension of the limit set of !; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of !. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its… Expand

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#### References

SHOWING 1-10 OF 19 REFERENCES

Lattices in rank one Lie groups over local fields

- Mathematics
- 1991

AbstractWe prove that if
$$G = \underline G (K)$$
is theK-rational points of aK-rank one semisimple group
$$\underline G $$
over a non archimedean local fieldK, thenG has cocompact non-arithmetic… Expand

Covering theory for graphs of groups

- Mathematics
- 1993

Abstract A tree action ( G, X ), consisting of a group G acting on a tree X , is encoded by a ‘quotient graph of groups’ A = G ⧹⧹ X . We introduce here the appropriate notion of morphism A → A′ = G… Expand

Schottky Groups and Mumford Curves

- Mathematics
- 1980

Discontinuous groups.- Mumford curves via automorphic forms.- The geometry of mumford curves.- Totally split curves and universal coverings.- Analytic reductions of algebraic curves.- Jacobian… Expand

Ergodic theory, symbolic dynamics, and hyperbolic spaces

- Mathematics
- 1991

Alan F. Beardon: An introduction to hyperbolic geometry Michael Keane: Ergodic theory and subshifts of finite type Anthony Manning: Dynamics of geodesic and horocycle flows on surfaces of constant… Expand

The density at infinity of a discrete group of hyperbolic motions

- Mathematics
- 1979

© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www.… Expand

Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits

- Mathematics
- 1989

0. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Renewal theorems in symbolic dynamics . . . . . . . . . . . . 1. Background: Shifts, suspension flows, thermodynamic… Expand

Actions quasi-convexes d'un groupe hyperbolique : flot géodésique

- Philosophy
- 1993

On etudie les actions isometriques quasi-convexes d'un groupe hyperbolique au sens de M. Gromov sur les espaces metriques simplement connexes a courbure strictement negative. A une telle action sont… Expand

Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov.

- Mathematics
- 1993

0. Introduction. Considerons Γespace hyperbolique usuel H, equipe d'un point base XQ . Munissons le bord dH de H de la mέtrique visuelle obtenue en regardant dH a partir de XQ (la distance entre deux… Expand

Geometrical methods of symbolic coding

- Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Eds T. Bedford, M. Keane and C. Series. Oxford University Press
- 1991

Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees (London Math

- Soc. Lecture Note Series 162). Cambridge University Press
- 1991