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

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.

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.

THE TRACE FUNCTION AND COMPLEX KLEINIAN GROUPS IN $P_{\mathbb{C}}^{2}$

2008 ◽  
Vol 19 (07) ◽  
pp. 865-890 ◽  
Author(s):  
JUAN-PABLO NAVARRETE
Keyword(s):  
Fixed Points ◽  
Kleinian Group ◽  
Cyclic Subgroup ◽  
Kleinian Groups ◽  
Trace Function ◽  
Limit Set

It is well known that the elements of PSL(2, ℂ) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in PU(2,1), then there are the equivalent notions of elliptic, parabolic or loxodromic elements; Goldman classified these transformations by their trace. In this work we extend the classification of elements of PU(2,1) to all of PSL(3, ℂ); we also extend to this setting the theorem that classifies them according to their trace. We use the notion of limit set introduced by Kulkarni, and calculate the limit set of every cyclic subgroup of PSL(3, ℂ) acting on [Formula: see text]. Given a classical Kleinian group it is possible to "suspend" this group to a subgroup of PSL(3, ℂ); we also calculate the limit set of this suspended group.

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.

scholarly journals Measurable rigidity for Kleinian groups

2015 ◽  
Vol 36 (8) ◽  
pp. 2498-2511
Author(s):  
WOOJIN JEON ◽  
KEN’ICHI OHSHIKA
Keyword(s):  
Kleinian Groups ◽  
Rigidity Theorem ◽  
Möbius Transformation ◽  
Limit Set ◽  
Boundary Map ◽  
Divergence Type ◽  
Mobius Transformation

Let$G,H$be two Kleinian groups with homeomorphic quotients$\mathbb{H}^{3}/G$and$\mathbb{H}^{3}/H$. We assume that$G$is of divergence type, and consider the Patterson–Sullivan measures of$G$and$H$. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map$\widehat{k}$from the limit set$\unicode[STIX]{x1D6EC}_{G}$of$G$to that of$H$is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such$\widehat{k}$always exists. In fact, we shall construct$\widehat{k}$concretely from the Cannon–Thurston maps of$G$and$H$.

The Limiting Behavior of Sequences of Quasiconformal Mappings

1990 ◽  
Vol 33 (4) ◽  
pp. 494-502
Author(s):  
Beat Aebischer
Keyword(s):  
Convergence Property ◽  
Quasiconformal Mappings ◽  
Kleinian Groups ◽  
Limit Set ◽  
Möbius Transformations ◽  
Limiting Behavior ◽  
Mobius Transformations

AbstractThe limiting behavior of sequences of quasiconformal homeomorphisms of the n-sphere Sn is studied using a substitute to the Poincaré extension of Möbius transformations introduced by Tukia. Adapted versions of the limit set and the conical limit set known in the theory of Kleinian groups are utilized. Most of the results also hold for families of homeomorphisms of Sn with the convergence property introduced by Gehring and Martin.

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.

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