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 MANIFOLDS THAT ARE NOT LEAVES OF CODIMENSION ONE FOLIATIONS

2013 ◽  
Vol 24 (14) ◽  
pp. 1350102 ◽  
Author(s):  
FÁBIO S. SOUZA ◽  
PAUL A. SCHWEITZER
Keyword(s):  
Compact Manifold ◽  
Homotopy Groups ◽  
Two Dimensional ◽  
Open Manifolds ◽  
Codimension One ◽  
Simply Connected Manifolds ◽  
Simply Connected

We present new open manifolds that are not homeomorphic to leaves of any C0 codimension one foliation of a compact manifold. Among them are simply connected manifolds of dimension d ≥ 5 that are non-periodic in homotopy, namely in their two-dimensional homotopy groups.

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.

CONNECTIVITY PROPERTIES OF SIERPIŃSKI RELATIVES

Fractals ◽  
2011 ◽  
Vol 19 (04) ◽  
pp. 481-506 ◽  
Author(s):  
T. D. TAYLOR
Keyword(s):  
Fractal Dimension ◽  
Multiply Connected ◽  
Simply Connected ◽  
Totally Disconnected

This paper presents a study of the connectivity of the class of fractals known as the Sierpiński relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.

scholarly journals On the structure of the family of Cherry fields on the torus

1985 ◽  
Vol 5 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Colin Boyd
Keyword(s):  
Vector Field ◽  
Hausdorff Dimension ◽  
Rotation Number ◽  
Vector Fields ◽  
Cantor Set ◽  
Irrational Rotation ◽  
Codimension One ◽  
The Family ◽  
Irrational Rotation Number

AbstractA class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.

scholarly journals Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

2018 ◽  
Vol 2020 (19) ◽  
pp. 6007-6041 ◽  
Author(s):  
Yuhei Suzuki
Keyword(s):  
Amenable Group ◽  
Cantor Set ◽  
Stable Rank ◽  
Crossed Product ◽  
Local Version ◽  
Crossed Products ◽  
Subexponential Growth ◽  
Minimal Action ◽  
New Strategy ◽  
Totally Disconnected

Abstract We extend Matui’s notion of almost finiteness to general étale groupoids and show that the reduced groupoid C$^{\ast }$-algebras of minimal almost finite groupoids have stable rank 1. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result: (1) for any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank 1; (2) any countable amenable group admits a minimal action on the Cantor set, all whose minimal extensions form the crossed product of stable rank 1; and (3) for any amenable group, the crossed product of the universal minimal action has stable rank 1.

Qualitative theory and expansion growth of transversely piecewise-smooth foliated ${S}^{1}$-bundles

1997 ◽  
Vol 17 (2) ◽  
pp. 331-347
Author(s):  
SHINJI EGASHIRA
Keyword(s):  
Compact Manifold ◽  
Qualitative Theory ◽  
Piecewise Smooth ◽  
Qualitative Properties ◽  
Codimension One

We study a qualitative theory of compact, transversely piecewise-smooth foliated $S^1$-bundles. We show that it has the same qualitative properties as that of smooth codimension-one foliations on a compact manifold. By the obtained qualitative properties, we can deduce that the expansion growth of this foliation is classified and that the entropy of this foliation is positive if and only if there exists a resilient leaf.

Analytic deformations of pencils and integrable one-forms having a first integral

2020 ◽  
Vol 31 (11) ◽  
pp. 2050089
Author(s):  
Bruno Scárdua
Keyword(s):  
Singular Point ◽  
First Integral ◽  
Product Formula ◽  
Analytic Family ◽  
Analytic Subset ◽  
Integrable Deformation ◽  
Codimension One ◽  
Dimension Formula ◽  
Geometric Conditions ◽  
Simply Connected

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family [Formula: see text] of integrable one-forms [Formula: see text] defined in a neighborhood [Formula: see text] of the initial singular point, and parametrized by the disc [Formula: see text]. The initial foliation is defined by [Formula: see text]. The second type, more restrictive, is given by an integrable holomorphic one-form [Formula: see text] defined in the product [Formula: see text]. Then, the initial foliation is defined by the slice restriction [Formula: see text]. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by [Formula: see text] for some germ of holomorphic function [Formula: see text] at the origin [Formula: see text]. We assume that the germ [Formula: see text] is irreducible and that the typical fiber of [Formula: see text] is simply-connected. This is the case if outside of a dimension [Formula: see text] analytic subset [Formula: see text], the analytic hypersurface [Formula: see text] has only normal crossings singularities. We then prove that, if cod sing [Formula: see text] then the (germ of the) developing foliation given by [Formula: see text] also exhibits a holomorphic first integral. For the general case, i.e. cod sing [Formula: see text], we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations [Formula: see text], of a local pencil [Formula: see text], for [Formula: see text]. For dimension [Formula: see text] we consider [Formula: see text]. For dimension [Formula: see text] we assume some generic geometric conditions on [Formula: see text] and [Formula: see text]. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type [Formula: see text] with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form [Formula: see text] with some additional properties, provided that for [Formula: see text] the axes remain invariant for the foliations [Formula: see text].

scholarly journals Shifts of finite type as fundamental objects in the theory of shadowing

2019 ◽  
Vol 220 (3) ◽  
pp. 715-736 ◽  
Author(s):  
Chris Good ◽  
Jonathan Meddaugh
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Inverse Limit ◽  
Cantor Set ◽  
Continuous Map ◽  
Fundamental Relationship ◽  
Directed System ◽  
Totally Disconnected

AbstractShifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and $$f:X\rightarrow X$$f:X→X a continuous map. We demonstrate that f has shadowing if and only if the system $$(f,X)$$(f,X) is (conjugate to) the inverse limit of a directed system satisfying the Mittag-Leffler condition and consisting of shifts of finite type. In particular, this implies that, in the case that X is the Cantor set, f has shadowing if and only if (f, X) is the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Moreover, in the general compact metric case, where X is not necessarily totally disconnected, we prove that f has shadowing if $$(f,X)$$(f,X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits.

scholarly journals Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

2012 ◽  
Vol 4 (2) ◽  
pp. 315
Author(s):  
N. Salma
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Groups ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Codimension One ◽  
Affine Groups ◽  
Simply Connected

Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989, 1996). On the other hand, an analogous criterion to Kobayashi’s equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let  be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups  and  are equivalent: (i) the action of  on  is proper; (ii)  is compact for any  The condition (i) is important in the study of discontinuous groups for the homogeneous space , while the second condition (ii) can easily be checked. The implication (i)  (ii) is obvious, and the opposite implication (ii)  (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ? (ii) for certain affine nilpotent Lie groups . Keywords: Affine nilpotent groups; Homogeneous manifolds; Proper actions; Properly discontinuous actions; Simply connected nilpotent Lie groups; Compact isotropy property (CI);  Eigenvalues. © 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v4i2.7889 J. Sci. Res. 4 (2), 315-326 (2012)

scholarly journals Parameter rigid actions of simply connected nilpotent Lie groups

2012 ◽  
Vol 33 (6) ◽  
pp. 1864-1875 ◽  
Author(s):  
HIROKAZU MARUHASHI
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Compact Manifold ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Dense Orbit ◽  
Simply Connected

AbstractWe show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

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