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

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 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.

Infinite iterated function systems with overlaps

2014 ◽  
Vol 36 (3) ◽  
pp. 890-907 ◽  
Author(s):  
SZE-MAN NGAI ◽  
JI-XI TONG
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Topological Pressure ◽  
Separation Condition ◽  
Limit Sets ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Weak Separation ◽  
Separation Conditions

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

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.

Limit Sets of Typical Homeomorphisms

2012 ◽  
Vol 55 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Nilson C. Bernardes
Keyword(s):  
Hausdorff Dimension ◽  
Borel Measure ◽  
Cantor Set ◽  
Limit Set ◽  
Positive Borel Measure ◽  
Dense Orbit ◽  
Limit Sets ◽  
Dimension Zero

AbstractGiven an integer n ≥ 3, a metrizable compact topological n-manifold X with boundary, and a finite positive Borel measure μ on X, we prove that for the typical homeomorphism f : X → X, it is true that for μ-almost every point x in X the limit set ω( f, x) is a Cantor set of Hausdorff dimension zero, each point of ω(f, x) has a dense orbit in ω(f, x), f is non-sensitive at each point of ω(f, x), and the function a → ω(f, a) is continuous at x.

Calculating Hausdorff dimension of Julia sets and Kleinian limit sets

2002 ◽  
Vol 124 (3) ◽  
pp. 495-545 ◽  
Author(s):  
Oliver Jenkinson ◽  
Mark Pollicott
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Limit Sets
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