Smale–Williams attractor

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

where each S_i is a circle and f_i is the map that uniformly wraps the circle S_i+1n_i times (n_i ≥ 2) around the circle S_i. 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.

In the special case where all n_i 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 Vietoris for n = 2 and by 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 dynamical systems.

Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in R³.

Fix a sequence of natural numbers {n_i}, n_i ≥ 2. Let T₀ = S¹ × D be a solid torus. For each i ≥ 0, choose a solid torus T_i+1 that is wrapped longitudinally n_i times inside the solid torus T_i. Then their intersection

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