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Solenoid (mathematics)

Class of compact connected topological spaces / From Wikipedia, the free encyclopedia

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In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms

This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.
The Smale-Williams solenoid.

where each is a circle and fi is the map that uniformly wraps the circle for times () around the circle . This construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a compact topological group.

Solenoids were first introduced by Vietoris for the case,[1] and by van Dantzig the case, where is fixed.[2] Such a solenoid arises as a one-dimensional expanding attractor, or Smale–Williams attractor, and forms an important example in the theory of hyperbolic dynamical systems.