scholarly journals A wild Cantor set as the limit set of a conformal group action on $S\sp 3$

1987 ◽  
Vol 99 (4) ◽  
pp. 623-623
Author(s):  
M. Bestvina ◽  
D. Cooper
Keyword(s):  
Group Action ◽  
Conformal Group ◽  
Cantor Set ◽  
Limit Set ◽  
Wild Cantor Set

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.

Schottky type groups and Kleinian groups acting on S3 with the limit set a wild Cantor set

1995 ◽  
Vol 26 (1) ◽  
pp. 1-45
Author(s):  
Ricardo Bianconi ◽  
Nikolay Gusevskii ◽  
Helen Klimenko
Keyword(s):  
Cantor Set ◽  
Kleinian Groups ◽  
Limit Set ◽  
Schottky Type ◽  
Wild Cantor 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.

Adding machines and wild attractors

1997 ◽  
Vol 17 (6) ◽  
pp. 1267-1287 ◽  
Author(s):  
HENK BRUIN ◽  
GERHARD KELLER ◽  
MATTHIAS ST. PIERRE
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Cantor Set ◽  
Limit Set ◽  
Unimodal Maps ◽  
Irrational Rotations ◽  
Circle Rotation ◽  
Omega Limit Set ◽  
Adding Machines

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

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.

scholarly journals On Limit Sets of Monotone Maps on Dendroids

2020 ◽  
Vol 5 (2) ◽  
pp. 311-316
Author(s):  
E.N. Makhrova
Keyword(s):  
Periodic Orbit ◽  
Periodic Points ◽  
Cantor Set ◽  
List Type ◽  
Limit Set ◽  
Limit Sets ◽  
Monotone Map ◽  
Monotone Maps

AbstractLet X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1)\omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ;(2)ω(x, f ) is either a periodic orbit or a minimal Cantor set.In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3)\Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f.The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

scholarly journals A wild Cantor set in the Hilbert cube

1968 ◽  
Vol 24 (1) ◽  
pp. 189-193 ◽  
Author(s):  
Raymond Wong
Keyword(s):  
Cantor Set ◽  
Hilbert Cube ◽  
Wild Cantor Set

Ends of Groups

2017 ◽  
Author(s):  
Nic Koban ◽  
John Meier
Keyword(s):  
Free Group ◽  
Group Action ◽  
Cantor Set ◽  
Braid Groups ◽  
Infinite Graphs ◽  
Research Projects ◽  
Semidirect Products ◽  
Number Of Ends ◽  
Ends Of Groups ◽  
Insight Into

This chapter focuses on the ends of a group. It first constructs a group action on the Cantor set and creates a free group from bijections of the Cantor set before showing how the idea of trying to understand what is happening at infinity for an infinite group is captured by the phrase “the ends of a group.” It then explores the notion of ends in the context of infinite graphs and presents examples that provide some insight into the number of ends of groups. It also looks at semidirect products and demonstrates how to calculate the number of ends of the braid groups before moving beyond the process of counting the ends of a group, taking into account the ends of the 4-valent tree. The discussion includes exercises and research projects.

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