Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in R3. Fix a sequence of natural numbers, ni ≥ 2. Let T0 = S1 × D be a solid torus. For each i ≥ 0, choose a solid torus Ti+1 that is wrapped longitudinally ni times inside the solid torus Ti. Then their intersection is homeomorphic to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence. Here is a variant of this construction isolated by Stephen Smale as an example of an expanding attractor in the theory of smooth dynamical systems. Denote the angular coordinate on the circle S1 by t and consider the complex coordinate z on the two-dimensional unit diskD. Let f be the map of the solid torus T = S1 × D into itself given by the explicit formula This map is a smooth embedding of T into itself that preserves the foliation by meridional disks. If T is imagined as a rubber tube, the map f stretches it in the longitudinal direction, contracts each meridional disk, and wraps the deformed tube twice inside T with twisting, but without self-intersections. The hyperbolic setΛ of the discrete dynamical system is the intersection of the sequence of nested solid tori described above, where Ti is the image of T under the ith iteration of the map f. This set is a one-dimensional attractor, and the dynamics of f on Λ has the following interesting properties:
meridional disks are the stable manifolds, each of which intersects Λ over a Cantor set
General theory of solenoids and expanding attractors, not necessarily one-dimensional, was developed by R. F. Williams and involves a projective system of infinitely many copies of a compact branched manifold in place of the circle, together with an expanding self-immersion.
Solenoids whose have the same value are known as -adic solenoids. A ring structure could be defined on in one of two ways: as the direct product of the circle ring and the ring of p-adic integers, or in a process similar to the algebraic construction of the p-adic numbers by inverse limits as follows: We start with the inverse limit of the rings : a -adic solenoid is then a sequence such that is in, and if, then Every real number when expressed in base defines such a sequence by and can therefore be regarded as a -adic solenoid. For example, in this case tau as a 2-adic solenoid would be written as the sequence. The operators of the ring amount to pointwise addition and multiplication of such sequences. This is well defined because addition and multiplication commute with the "" operator; see modular arithmetic. As a result, -adic solenoids have a base doubly infinitepositional notation representation, similar to p-adic integers or real numbers.
Profinite real numbers
A profinite real number is an element of the ring where indicates the profinite completion of, the index p runs over all prime numbers, and is the p-adic solenoid.
