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
 $f_{i}:S_{i+1}\to S_{i}\quad \forall i\geq 0$
 This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.
Algebraic structure → Group theory Group theory  



Modular groups


Infinite dimensional Lie group


where each $S_{i}$ is a circle and f_{i} is the map that uniformly wraps the circle $S_{i+1}$ for $n_{i+1}$ times ($n_{i+1}\geq 2$) around the circle $S_{i}$. This construction can be carried out geometrically in the threedimensional Euclidean space R^{3}. A solenoid is a onedimensional homogeneous indecomposable continuum that has the structure of a compact topological group.
Solenoids were first introduced by Vietoris for the $n_{i}=2$ case,[1] and by van Dantzig the $n_{i}=n$ case, where $n\geq 2$ is fixed.[2] Such a solenoid arises as a onedimensional expanding attractor, or Smale–Williams attractor, and forms an important example in the theory of hyperbolic dynamical systems.