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In mathematics, a solenoid is a compact connected topological space that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms
where each S i {\displaystyle S_{i}} is a circle and fi is the map that uniformly wraps the circle S i + 1 {\displaystyle S_{i+1}} for n i + 1 {\displaystyle n_{i+1}} times around the circle S i {\displaystyle S_{i}}. This construction can be carried out geometrically in the three-dimensional Euclidean space R. 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 n i = 2 {\displaystyle n_{i}=2} case, and by van Dantzig the n i = n {\displaystyle n_{i}=n} case, where n ≥ 2 {\displaystyle n\geq 2} is fixed. 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.