scholarly journals Dynamics of Kleinian groups—the Hausdorff dimension of limit sets

2001 ◽  
pp. 23-44 ◽  
Author(s):  
Katsuhiko Matsuzaki
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Groups ◽  
Limit Sets

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

Discrete Lyapunov exponents and Hausdorff dimension

2000 ◽  
Vol 20 (1) ◽  
pp. 145-172 ◽  
Author(s):  
SHMUEL FRIEDLAND
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Thermodynamic Formalism ◽  
Sufficient Conditions ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Subshifts Of Finite Type ◽  
Discrete Analogs

We study certain metrics on subshifts of finite type for which we define the discrete analogs of Lyapunov exponents. We prove Young's formula for $\mu$-Hausdorff dimension. We give sufficient conditions on the above metrics for which the Hausdorff dimension is given by thermodynamic formalism. We apply these results to the Hausdorff dimension of the limit sets of geometrically finite, purely loxodromic, Kleinian groups.

Diophantine approximation in Kleinian groups

1994 ◽  
Vol 116 (1) ◽  
pp. 57-78 ◽  
Author(s):  
Bernd Stratmann
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Detailed Understanding ◽  
Diophantine Approximations ◽  
Natural Way

AbstractThe δ-homogeneity of the Patterson measure is used for a closer study of the limit sets of Kleinian groups. A combination of the properties of this measure with concepts of diophantine approximations is shown to lead to a more detailed understanding of these limit sets. In particular, it is seen to how great an extent the studies of these sets, in terms of Hausdorff measure or Hausdorff dimension, are limited in a natural way.

The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups

2000 ◽  
Vol 128 (1) ◽  
pp. 123-139 ◽  
Author(s):  
KATSUHIKO MATSUZAKI
Keyword(s):  
Hausdorff Dimension ◽  
Discrete Subgroup ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Full Dimension

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

Length distortion and the Hausdorff dimension of limit sets

2000 ◽  
Vol 122 (3) ◽  
pp. 465-482 ◽  
Author(s):  
Martin Bridgeman ◽  
Edward C. Taylor
Keyword(s):  
Hausdorff Dimension ◽  
Limit Sets ◽  
Length Distortion

FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

1995 ◽  
Vol 06 (01) ◽  
pp. 19-32 ◽  
Author(s):  
NIKOLAY GUSEVSKII ◽  
HELEN KLIMENKO
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Wild Cantor Set ◽  
Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

Sign in / Sign up

Export Citation Format

Share Document