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.

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.

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.

scholarly journals Limit sets of geometrically finite free Kleinian groups

1984 ◽  
Vol 36 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Tohru Akaza ◽  
Katsumi Inoue
Keyword(s):  
Kleinian Groups ◽  
Limit Sets ◽  
Geometrically Finite

scholarly journals Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind

2015 ◽  
Vol 178 (1) ◽  
pp. 95-101
Author(s):  
Lior Fishman ◽  
David Simmons ◽  
Mariusz Urbański
Keyword(s):  
Kleinian Groups ◽  
Limit Sets ◽  
Geometrically Finite

scholarly journals Micromeasure distributions and applications for conformally generated fractals

2015 ◽  
Vol 159 (3) ◽  
pp. 547-566 ◽  
Author(s):  
JONATHAN M. FRASER ◽  
MARK POLLICOTT
Keyword(s):  
Finite Type ◽  
Julia Sets ◽  
Gibbs Measures ◽  
Schottky Groups ◽  
Limit Sets ◽  
Distance Set ◽  
Self Similar ◽  
Subshifts Of Finite Type

AbstractWe study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

THERMODYNAMIC FORMALISM FOR RANDOM TRANSFORMATIONS REVISITED

2008 ◽  
Vol 08 (01) ◽  
pp. 77-102 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Finite Type ◽  
Type Theorem ◽  
Thermodynamic Formalism ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Topological Mixing ◽  
Subshifts Of Finite Type

We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron–Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.

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.

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