Hausdorff dimension of a limit set for a family of nonholomorphic perturbations of the mapztoz2

Nonlinearity ◽  
1999 ◽  
Vol 12 (5) ◽  
pp. 1439-1448 ◽  
Author(s):  
J J Szczyrek
Keyword(s):  
Hausdorff Dimension ◽  
Limit Set

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.

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.

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).

A NOTE ON THE UNIMODAL FEIGENBAUM'S MAPS

2009 ◽  
Vol 23 (14) ◽  
pp. 3101-3111
Author(s):  
GUIFENG HUANG ◽  
LIDONG WANG ◽  
GONGFU LIAO
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Periodic Points ◽  
Limit Set ◽  
Almost Periodic ◽  
Limit Sets ◽  
Shift Map ◽  
Renormalization Operator

We mainly investigate the likely limit sets and the kneading sequences of unimodal Feigenbaum's maps (Feigenbaum's map can be regarded as the fixed point of the renormalization operator [Formula: see text], where λ is to be determined). First, we estimate the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum's map and then for every decimal s ∈ (0, 1), we construct a unimodal Feigenbaum's map which has a likely limit set with Hausdorff dimension s. Second, we prove that the kneading sequences of unimodal Feigenbaum's maps are uniformly almost periodic points of the shift map but not periodic ones.

scholarly journals Uniform congruence counting for Schottky semigroups in SL2(𝐙)

2019 ◽  
Vol 2019 (753) ◽  
pp. 89-135 ◽  
Author(s):  
Michael Magee ◽  
Hee Oh ◽  
Dale Winter
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Continued Fractions ◽  
Euclidean Norm ◽  
Limit Set ◽  
The Family ◽  
Prime Factors ◽  
Uniform Congruence

AbstractLet Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})}, and for {q\in\mathbf{N}}, let{\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}}be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R: for all positive integer q with no small prime factors,\#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(% \mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon})as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})}, which arises in the study of Zaremba’s conjecture on continued fractions.

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.

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