scholarly journals A wild Cantor set in $E^n$ with simply connected complement

1974 ◽  
Vol 86 (1) ◽  
pp. 9-27 ◽  
Author(s):  
D. DeGryse ◽  
R. Osborne
Keyword(s):  
Cantor Set ◽  
Simply Connected ◽  
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 Exotic Nonleaves with Infinitely Many Ends

2021 ◽  
Author(s):  
Carlos Meniño Cotón ◽  
Paul A Schweitzer
Keyword(s):  
Compact Manifold ◽  
Cantor Set ◽  
Remarkable Case ◽  
Realization Problem ◽  
Codimension One ◽  
Simply Connected ◽  
New Criterion ◽  
Totally Disconnected

Abstract We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $\Lambda $ admits a continuum of smoothings, which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are “exotic”, that is, homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold, being the 1st examples in this class.

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

THE SETS OF POINTS WITH BOUNDED ORBITS FOR GENERALIZED CHEBYSHEV MAPPINGS

2001 ◽  
Vol 11 (01) ◽  
pp. 91-107
Author(s):  
KEISUKE UCHIMURA
Keyword(s):  
Dynamical Systems ◽  
Projective Space ◽  
Complex Variable ◽  
Complex Projective Space ◽  
Cantor Set ◽  
Quadratic Polynomials ◽  
Holomorphic Map ◽  
Simply Connected ◽  
Complex Projective ◽  
Set Of Points

We study the dynamical systems given by generalized Chebyshev mappings [Formula: see text] and show that (1) the set of points with bounded orbits of Fc(z) is connected and its complement in C∪{∞} is simply connected if and only if -4 ≤ c ≤ 2; (2) if c > 2, then the set of points with bounded orbits of Fc(z) is Cantor set. These results are the analogue of the theory of filled Julia sets of quadratic polynomials in one complex variable. We show that the mapping Fc(z) has relation to an important holomorphic map on the complex projective space P2.

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

EXAMPLES OF USING BINARY CANTOR SETS TO STUDY THE CONNECTIVITY OF SIERPIŃSKI RELATIVES

Fractals ◽  
2012 ◽  
Vol 20 (01) ◽  
pp. 61-75 ◽  
Author(s):  
T. D. TAYLOR ◽  
C. HUDSON ◽  
A. ANDERSON
Keyword(s):  
Fractal Dimension ◽  
Sierpinski Gasket ◽  
Cantor Set ◽  
Cantor Sets ◽  
Sierpiński Gasket ◽  
Multiply Connected ◽  
Simply Connected ◽  
Totally Disconnected

The Sierpiński relatives form a class of fractals that all have the same fractal dimension, but different topologies. This class includes the well-known Sierpiński gasket. Some relatives are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. This paper presents examples of relatives for which binary Cantor sets are relevant for the connectivity. These Cantor sets are variations of the usual middle thirds Cantor set, and their binary descriptions greatly aid in the determination of the connectivity of the corresponding relatives.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Exploring the landscape of CHL strings on Td

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Anamaría Font ◽  
Bernardo Fraiman ◽  
Mariana Graña ◽  
Carmen A. Núñez ◽  
Héctor Parra De Freitas
Keyword(s):  
Gauge Group ◽  
Moduli Space ◽  
Dynkin Diagram ◽  
Fundamental Groups ◽  
Computer Algorithms ◽  
Abelian Gauge ◽  
Reduced Rank ◽  
Systematic Exploration ◽  
String Models ◽  
Simply Connected

Abstract Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2.

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