Solenoid (mathematics)

This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.

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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

${\displaystyle f_{i}:S_{i+1}\to S_{i}\quad \forall i\geq 0}$

where each ${\displaystyle S_{i}}$ is a circle and f_i is the map that uniformly wraps the circle ${\displaystyle S_{i+1}}$ for ${\displaystyle n_{i+1}}$ times (${\displaystyle n_{i+1}\geq 2}$) around the circle ${\displaystyle S_{i}}$.^{[1]: Ch. 2 Def. (10.12)} 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 an abelian compact topological group.

Solenoids were first introduced by Vietoris for the ${\displaystyle n_{i}=2}$ case,^[2] and by van Dantzig the ${\displaystyle n_{i}=n}$ case, where ${\displaystyle n\geq 2}$ is fixed.^[3] 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.