scholarly journals The critical exponent, the Hausdorff dimension of the limit set and the convex core entropy of a Kleinian group

2015 ◽  
Vol 19 (8) ◽  
pp. 159-196
Author(s):  
Kurt Falk ◽  
Katsuhiko Matsuzaki
Keyword(s):  
Hausdorff Dimension ◽  
Critical Exponent ◽  
Kleinian Group ◽  
Limit Set ◽  
Convex Core

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.

A variation formula for the topological entropy of convex-cocompact manifolds

2010 ◽  
Vol 31 (6) ◽  
pp. 1849-1864 ◽  
Author(s):  
SAMUEL TAPIE
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Formula ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Kleinian Group ◽  
Limit Set ◽  
Analytic Family ◽  
Variation Formula ◽  
Sectional Curvatures ◽  
Convex Cocompact

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

scholarly journals The fractal structure of limit set of solution space of a doubly periodic Ricatti equation

1997 ◽  
Vol 20 (4) ◽  
pp. 707-712 ◽  
Author(s):  
Ke-Ying Guan
Keyword(s):  
Hausdorff Dimension ◽  
Riccati Equation ◽  
Kleinian Group ◽  
Fractal Structure ◽  
Solution Space ◽  
Limit Set ◽  
Ricatti Equation

The limit set of the Kleinian group of a given doubly periodic Riccati equation is proved to have a fractal structure if the parameterδ(λ)of the equation is greater than3+22, and a possible Hausdorff dimension is suggested to the limit set.

scholarly journals Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry

2019 ◽  
Author(s):  
Olivier Glorieux ◽  
Daniel Monclair
Keyword(s):  
Hausdorff Dimension ◽  
Critical Exponent ◽  
Hyperbolic Geometry ◽  
Limit Set ◽  
Rigidity Result ◽  
Famous Theorem ◽  
Limit Sets ◽  
Convex Cocompact

AbstractThe aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\textrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud, and Kassel, called ${\mathbb{H}}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in ${\mathbb{H}}^{2,1}={\mathbb{A}}\textrm{d}{\mathbb{S}}^3$, which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$D hyperbolic geometry.

scholarly journals Conformality for a robust class of non-conformal attractors

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria Beatrice Pozzetti ◽  
Andrés Sambarino ◽  
Anna Wienhard
Keyword(s):  
Hausdorff Dimension ◽  
Critical Exponent ◽  
Convergence Property ◽  
Limit Set ◽  
Limit Sets ◽  
Differentiability Property ◽  
Anosov Representations

AbstractIn this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent.

scholarly journals Surface groups acting on spaces

2017 ◽  
Vol 39 (7) ◽  
pp. 1843-1856
Author(s):  
GEORGIOS DASKALOPOULOS ◽  
CHIKAKO MESE ◽  
ANDREW SANDERS ◽  
ALINA VDOVINA
Keyword(s):  
Hausdorff Dimension ◽  
Group Action ◽  
Metric Space ◽  
Hyperbolic Plane ◽  
Harmonic Map ◽  
Surface Group ◽  
Surface Groups ◽  
Limit Set ◽  
Convex Cocompact

Harmonic map theory is used to show that a convex cocompact surface group action on a $\text{CAT}(-1)$ metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a $\text{CAT}(-1)$ space has Hausdorff dimension $\geq 1$, where the inequality is strict unless the action is Fuchsian.

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