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:''This page discusses a class of topological groups. For the wrapped loop of wire, see [[Solenoid]].''
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[[File:Smale-Williams Solenoid Large.png|thumb|right|300px|The Smale-Williams solenoid.]]
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In [[mathematics]], a '''solenoid''' is a [[compact space|compact]] [[connected space|connected]] [[topological space]] (i.e. a [[continuum (topology)|continuum]]) that may be obtained as the [[inverse limit]] of an inverse system of [[topological group]]s and [[continuous function|continuous]] [[homomorphism]]s 


:(''S''<sub>''i''</sub>, ''f''<sub>''i''</sub>), &nbsp;&nbsp;&nbsp; ''f''<sub>''i''</sub>: ''S''<sub>''i''+1</sub> &rarr; ''S''<sub>''i''</sub>, &nbsp;&nbsp;&nbsp; ''i'' &ge; 0,
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where each ''S<sub>i</sub>'' is a [[circle]] and ''f''<sub>''i''</sub> is the map that uniformly wraps the circle ''S''<sub>''i''+1</sub> ''n''<sub>''i''</sub> times (''n''<sub>''i''</sub> &ge; 2) around the circle ''S''<sub>''i''</sub>. This construction can be carried out geometrically in the three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>. A solenoid is a one-dimensional homogeneous [[indecomposable continuum]] that has the structure of a compact [[topological group]].  
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In the special case where all ''n''<sub>''i''</sub> have the same value ''n'', so that the inverse system is determined by the multiplication by ''n'' self map of the circle, solenoids were first introduced by [[Leopold Vietoris|Vietoris]] for ''n'' = 2 and by [[David van Dantzig|van Dantzig]] for an arbitrary ''n''. Such a solenoid arises as a one-dimensional '''expanding attractor''', or '''Smale–Williams attractor''', and forms an important example in the theory of [[hyperbolic dynamics|hyperbolic]] [[dynamical system]]s.
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== Geometric construction and the Smale–Williams attractor ==
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[[File:Smale-Williams Solenoid.png|thumb|300px|A solid torus wrapped twice around inside another solid torus in '''R'''<sup>3</sup>]]
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[[File:Solenoid.gif|thumb|300px|The first six steps in the construction of the Smale-Williams attractor.]]
Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in '''R'''<sup>3</sup>.


Fix a sequence of natural numbers {''n''<sub>''i''</sub>}, ''n''<sub>''i''</sub> &ge; 2. Let ''T''<sub>0</sub> = ''S''<sup>1</sup> &times; ''D'' be a [[solid torus]]. For each ''i'' &ge; 0, choose a solid torus ''T''<sub>''i''+1</sub> that is wrapped longitudinally ''n''<sub>''i''</sub> times inside the solid torus ''T''<sub>''i''</sub>. Then their intersection
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: <math>\Lambda=\bigcap_{i\ge 0}T_i</math>
 
is [[homeomorphic]] to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence {''n''<sub>''i''</sub>}.
 
Here is a variant of this construction isolated by [[Stephen Smale]] as an example of an '''expanding attractor''' in the theory of smooth dynamical systems. Denote the angular coordinate on the circle ''S''<sup>1</sup> by ''t'' (it is defined mod 2&pi;) and consider the complex coordinate ''z'' on the two-dimensional [[unit disk]] ''D''. Let ''f'' be the map of the solid torus ''T'' = ''S''<sup>1</sup> &times; ''D'' into itself given by the explicit formula
 
: <math> f(t,z) = \left(2t, \tfrac{1}{4}z + \tfrac{1}{2}e^{it}\right).</math>
 
This map is a smooth [[embedding]] of ''T'' into itself that preserves the [[foliation]] by meridional disks (the constants 1/2 and 1/4 are somewhat arbitrary, but it is essential that 1/4 < 1/2 and 1/4 + 1/2 < 1). If ''T'' is imagined as a rubber tube, the map ''f'' stretches it in the longitudinal direction, contracts each meridional disk, and wraps the deformed tube twice inside ''T'' with twisting, but without self-intersections. The [[hyperbolic set]] ''&Lambda;'' of the discrete dynamical system (''T'', ''f'') is the intersection of the sequence of nested solid tori described above, where ''T''<sub>''i''</sub> is the image of ''T'' under the ''i''th iteration of the map ''f''. This set is a one-dimensional (in the sense of [[topological dimension]]) [[attractor]], and the dynamics of ''f'' on ''&Lambda;'' has the following interesting properties:
 
* meridional disks are the [[stable manifold]]s, each of which intersects ''&Lambda;'' over a [[Cantor set]]
* [[periodic point]]s of ''f'' are [[dense subset|dense]] in ''&Lambda;''
* the map ''f'' is [[topologically transitive]] on ''&Lambda;''
 
General theory of solenoids and expanding attractors, not necessarily one-dimensional, was developed by R. F. Williams and involves a projective system of infinitely many copies of a compact [[branched manifold]] in place of the circle, together with an expanding self-[[immersion (mathematics)|immersion]].
 
== Pathological properties ==
 
Solenoids are [[compact space|compact]] [[metrizable space]]s that are [[connected space|connected]], but not [[locally connected]] or [[path connected]]. This is reflected in their [[pathological (mathematics)|pathological]] behavior with respect to various [[homology theories]], in contrast with the standard properties of homology for [[simplicial complex]]es. In [[Čech cohomology|Čech homology]], one can construct a non-exact [[long exact sequence|long homology sequence]] using a solenoid. In [[Steenrod]]-style homology theories, the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space.
 
==See also==
*[[Protorus]], a class of topological groups that includes the solenoids
 
== References ==
{{No footnotes|date=January 2012}}
 
* D. van Dantzig, ''Ueber topologisch homogene Kontinua'',  Fund. Math. 15  (1930), pp. 102–125
* {{eom|title=Solenoid|id=S/s086040}}
* Clark Robinson, ''Dynamical systems: Stability, Symbolic Dynamics and Chaos'', 2nd edition, CRC Press, 1998 ISBN 978-0-8493-8495-0
* S. Smale, [http://www.ams.org/bull/1967-73-06/S0002-9904-1967-11798-1/S0002-9904-1967-11798-1.pdf ''Differentiable dynamical systems''], [[Bulletin of the American Mathematical Society|Bull. of the AMS]], 73 (1967), 747 &ndash; 817.  
*  L. Vietoris, ''Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen'', Math. Ann. 97 (1927), pp. 454–472
* Robert F. Williams, [http://www.numdam.org/item?id=PMIHES_1974__43__169_0 ''Expanding attractors''], Publ. Math. IHES, t. 43 (1974), p. 169–203
 
==Further reading==
*{{Citation |last=Semmes |first=Stephen |authorlink=Stephen Semmes |date=12 January 2012 |title=Some remarks about solenoids |arxiv=1201.2647}}
 
{{DEFAULTSORT:Solenoid (Mathematics)}}
[[Category:Topological groups]]
[[Category:Continuum theory]]
[[Category:Algebraic topology]]

Revision as of 18:28, 10 February 2014

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