TOPOLOGICAL DYNAMICS OF STABLE GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1199-1223 ◽  
Author(s):  
LUDOMIR NEWELSKI
Keyword(s):  
Inverse Limit ◽  
Topological Dynamics ◽  
Maximal Subgroups ◽  
Inverse Limits ◽  
Stable Theory ◽  
Definable Quotients

AbstractAssumeGis a group definable in a modelMof a stable theoryT. We prove that the semigroupSG(M) of completeG-types overMis an inverse limit of some semigroups type-definable inMeq. We prove that the maximal subgroups ofSG(M) are inverse limits of some definable quotients of subgroups ofG. We consider the powers of types in the semigroupSG(M) and prove that in a way every type inSG(M) is profinitely many steps away from a type in a subgroup ofSG(M).

Profinite Modules

1973 ◽  
Vol 16 (3) ◽  
pp. 405-415
Author(s):  
Gerard Elie Cohen
Keyword(s):  
Finite Group ◽  
General Theory ◽  
Finite Length ◽  
Finite Groups ◽  
Inverse Limit ◽  
Abelian Groups ◽  
Point Of View ◽  
Inverse Limits ◽  
Profinite Groups ◽  
Inverse Systems

An inverse limit of finite groups has been called in the literature a pro-finite group and we have extensive studies of profinite groups from the cohomological point of view by J. P. Serre. The general theory of non-abelian modules has not yet been developed and therefore we consider a generalization of profinite abelian groups. We study inverse systems of discrete finite length R-modules. Profinite modules are inverse limits of discrete finite length R-modules with the inverse limit topology.

On inverse limits of monounary algebras

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
Emília Halušková
Keyword(s):  
Inverse Limit ◽  
Inverse Limits

AbstractWe study inverse limits of monounary algebras. All monounary algebras A such that A can arise from A only by an inverse limit construction are described. We deal with an existence of an inverse limit. Some inverse limit closed classes are described. The paper ends with two problems.

A Note on Inverse Limits of Finite Spaces

1970 ◽  
Vol 13 (1) ◽  
pp. 69-70
Author(s):  
S. B. Nadler
Keyword(s):  
Finite Groups ◽  
Inverse Limit ◽  
Inverse System ◽  
Inverse Limits ◽  
Hausdorff Spaces ◽  
Finite Spaces

The following lemma, which appears as Lemma 4 in [5], was used to determine certain multicoherence properties of inverse limits of continua.Lemma. Let X denote the inverse limit of an inverse system {Xλ, fλμ, Λ} of compact Hausdorff spaces Xλ. If Xλ has no more than k components (where k < ∞ is fixed) for each λ ∊ Λ, then X has no more than k components.In this paper we give a set theoretic analogue of this lemma and an extension which was suggested to the author by Professor F. W. Lawvere. An application to inverse limits of finite groups is then given.

Dynamical properties of shift maps on inverse limits with a set valued function

2016 ◽  
Vol 38 (4) ◽  
pp. 1499-1524 ◽  
Author(s):  
JUDY KENNEDY ◽  
VAN NALL
Keyword(s):  
Topological Structure ◽  
Inverse Limit ◽  
Natural Generalization ◽  
Continuous Functions ◽  
Invariant Sets ◽  
Inverse Limits ◽  
Limit Space ◽  
Shift Map ◽  
Single Well ◽  
Inverse Limit Space

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

CARDINALITY OF INVERSE LIMITS WITH UPPER SEMICONTINUOUS BONDING FUNCTIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 167-174 ◽  
Author(s):  
MATEJ ROŠKARIČ ◽  
NIKO TRATNIK
Keyword(s):  
Recent Result ◽  
Inverse Limit ◽  
Inverse Limits ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Function

AbstractWe explore the cardinality of generalised inverse limits. Among other things, we show that, for any $n\in \{&#x2135;_{0},c,1,2,3,\dots \}$, there is an upper semicontinuous function with the inverse limit having exactly $n$ points. We also prove that if $f$ is an upper semicontinuous function whose graph is a continuum, then the cardinality of the corresponding inverse limit is either 1, $&#x2135;_{0}$ or $c$. This generalises the recent result of I. Banič and J. Kennedy, which claims that the same is true in the case where the graph is an arc.

scholarly journals Inverse limits of finite rank free groups

2012 ◽  
Vol 15 (6) ◽  
Author(s):  
Gregory R. Conner ◽  
Curtis Kent
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Finite Rank ◽  
Inverse Limit ◽  
Free Groups ◽  
Inverse System ◽  
Inverse Limits ◽  
Universal Group ◽  
Hawaiian Earring ◽  
Constant System

Abstract.We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group.

scholarly journals An inverse limit representation for complete Boolean algebras

1972 ◽  
Vol 13 (2) ◽  
pp. 164-166 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Inverse Limit ◽  
Direct Limit ◽  
Boolean Algebras ◽  
Inverse Limits ◽  
Finitely Generated ◽  
Systematic Account ◽  
Actual Fact ◽  
Infinite Lattices

There are two well-known methods to build up algebras from given algebras, the direct and inverse limits, and a systematic account of these constructions may be found in [2]. It is known that every algebra can be represented as a direct limit of finitely generated algebras although in some cases the representation is trivial. Furthermore, Haimo [3] has established a certain inverse limit representation for the class of all infinite Boolean algebras which generalises, in actual fact, to the class of all infinite lattices with 1. The purpose of this note is to exhibit a certain nontrivial inverse limit representation which is peculiar to the class of infinite, complete Boolean algebras.

scholarly journals The nub of an automorphism of a totally disconnected, locally compact group

2013 ◽  
Vol 34 (4) ◽  
pp. 1365-1394 ◽  
Author(s):  
GEORGE A. WILLIS
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Finite Depth ◽  
Locally Compact Group ◽  
Topological Dynamics ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Contraction Group ◽  
Totally Disconnected

AbstractTo any automorphism,$\alpha $, of a totally disconnected, locally compact group,$G$, there is associated a compact,$\alpha $-stable subgroup of$G$, here called thenubof$\alpha $, on which the action of$\alpha $is ergodic. Ergodic actions of automorphisms of compact groups have been studied extensively in topological dynamics and results obtained transfer, via the nub, to the study of automorphisms of general locally compact groups. A new proof that the contraction group of$\alpha $is dense in the nub is given, but it is seen that the two-sided contraction group need not be dense. It is also shown that each pair$(G, \alpha )$, with$G$compact and$\alpha $ergodic, is an inverse limit of pairs that have ‘finite depth’ and that analogues of the Schreier refinement and Jordan–Hölder theorems hold for pairs with finite depth.

scholarly journals Periodic points of solenoidal automorphisms in terms of inverse limits

2021 ◽  
Vol 22 (2) ◽  
pp. 321
Author(s):  
Sharan Gopal ◽  
Faiz Imam
Keyword(s):  
Inverse Limit ◽  
Periodic Points ◽  
Inverse Limits ◽  
One Dimensional ◽  
Higher Dimensional

<p>In this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, lim←k (S 1 , γk) of a sequence (γk) of maps on the circle S 1 . The periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also.</p>

scholarly journals Topological n-cells and Hilbert cubes in inverse limits

2018 ◽  
Vol 19 (1) ◽  
pp. 9
Author(s):  
Leonard R. Rubin
Keyword(s):  
Set Theory ◽  
Weak Topology ◽  
Inverse Limit ◽  
Metrizable Space ◽  
Inverse System ◽  
Inverse Limits ◽  
Hilbert Cube ◽  
Compact Metrizable Space ◽  
Metric Topology

<p>It has been shown by S. Mardešić that if a compact metrizable space X has dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then X must contain an arc.  We are going  to prove that  if X = (|K<sub>a</sub>|,p<sup>b</sup><sub>a</sub>,(A,)<a href="http://www.codecogs.com/eqnedit.php?latex=\preceq" target="_blank"><img title="\preceq" src="http://latex.codecogs.com/gif.latex?\preceq" alt="" /></a>)is an inverse system in set theory of triangulated polyhedra|K<sub>a</sub>|with simplicial  bonding  functions p<sup>b</sup><sub>a</sub> and X = lim X,  then  there  exists  a uniquely determined sub-inverse system X<sub>X</sub>= (|L<sub>a</sub>|, p<sup>b</sup><sub>a</sub>|L<sub>b</sub>|,(A,<a href="http://www.codecogs.com/eqnedit.php?latex=\preceq" target="_blank"><img title="\preceq" src="http://latex.codecogs.com/gif.latex?\preceq" alt="" /></a>)) of X where for each a, L<sub>a</sub> is a subcomplex of K<sub>a</sub>, each p<sup>b</sup><sub>a</sub>|L<sub>b</sub>|:|L<sub>b</sub>| → |L<sub>a</sub>| is  surjective,  and lim X<sub>X</sub> = X. We shall use this to generalize the Mardešić result by characterizing when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell and do the same in the case of an inverse system of finite triangulated polyhedra with simplicial bonding maps. We shall also characterize when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain an embedded copy of the Hilbert cube. In each of the above settings, all the polyhedra have the weak topology or all have the metric topology(these topologies being identical when the polyhedra are finite).</p>

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