scholarly journals The Hausdorff dimension of the limit set of a geometrically finite Kleinian group

1984 ◽  
Vol 152 (0) ◽  
pp. 127-140 ◽  
Author(s):  
Pekka Tukia
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Limit Set ◽  
Geometrically Finite ◽  
Finite Kleinian Group

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.

The Hausdorff dimension of bounded geodesics on geometrically finite manifolds

1997 ◽  
Vol 17 (1) ◽  
pp. 227-246 ◽  
Author(s):  
B. STRATMANN
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Kleinian Group ◽  
Negative Curvature ◽  
Exponent Of Convergence ◽  
Constant Negative Curvature ◽  
Geometrically Finite ◽  
Limit Points ◽  
Badly Approximable

In this paper we study the set of bounded geodesics on a general, geometrically finite $(N+1)$-manifold of constant negative curvature. We obtain the result that the Hausdorff dimension of this set is equal to $2 \delta$, where $\delta$ denotes the exponent of convergence of the associated Kleinian group. The proof of this shows, in particular, that if the group has parabolic elements, then the set of limit points which are badly approximable with respect to the parabolic fixed points has Hausdorff dimension equal to $\delta$.

scholarly journals Geometrically finite kleinian groups and parabolic elements

1998 ◽  
Vol 41 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Ken'ichi Ohshika
Keyword(s):  
Kleinian Group ◽  
Kleinian Groups ◽  
Deformation Space ◽  
Ideal Boundary ◽  
Conformal Deformation ◽  
Closed Curves ◽  
Geometrically Finite ◽  
Finite Kleinian Group ◽  
Torsion Free ◽  
The Ideal

Let Γ be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of Γ can be simultaneously made parabolic in a group on the boundary of the quasi-conformal deformation space of Γ. We shall prove that for this, it is sufficient that the classes of the system are represented by disjoint primitive simple closed curves on the ideal boundary of H3/Γ.

Equidistribution of parabolic fixed points in the limit set of Kleinian groups

1999 ◽  
Vol 19 (6) ◽  
pp. 1437-1484 ◽  
Author(s):  
SALVATORE COSENTINO
Keyword(s):  
Fixed Point ◽  
Riemann Hypothesis ◽  
Asymptotic Estimate ◽  
Counting Function ◽  
Weak Limit ◽  
Kleinian Groups ◽  
Limit Set ◽  
Geometrically Finite ◽  
Finite Kleinian Group ◽  
Parabolic Fixed Point

We show that the Patterson–Sullivan measure on the limit set of a geometrically finite Kleinian group with cusps can be recovered as a weak limit of sums of Dirac masses placed on an appropriate orbit of each parabolic fixed point. A corollary is a sharp asymptotic estimate for a natural counting function associated to a cuspidal subgroup. We also discuss the connection between the above counting and the Riemann hypothesis in some examples of arithmetical lattices.

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