Solenoid (mathematics)

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

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

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.

.

Construction

Geometric construction and the Smale–Williams attractor

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

${\displaystyle \Lambda =\bigcap _{i\geq 0}T_{i}}$

is

to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence {n_i}.

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 (it is defined mod 2π) and consider the complex coordinate z on the two-dimensional unit disk D. Let f be the map of the solid torus T = S¹ × D into itself given by the explicit formula

${\displaystyle f(t,z)=\left(2t,{\tfrac {1}{4}}z+{\tfrac {1}{2}}e^{it}\right).}$

This map is a smooth

, 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
• dense
  in Λ
• the map f is
  topologically transitive
  on Λ

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.

Construction in toroidal coordinates

In the toroidal coordinates with radius ${\displaystyle R}$, the solenoid can be parametrized by ${\displaystyle t\in \mathbb {R} }$ as

where

${\displaystyle \omega _{k}={\frac {1}{n_{1}\cdots n_{k}}},\quad r_{k}=R\delta _{1}\cdots \delta _{k}}$

Here, ${\displaystyle \delta _{k}}$ are adjustable shape-parameters, with constraint ${\displaystyle 0<\delta <1-{\frac {1}{1+\sin {\frac {\pi }{n_{k}}}}}}$. In particular, ${\displaystyle \delta ={\frac {1}{2n_{k}}}}$ works.

Let ${\displaystyle S\subset \mathbb {R} ^{3}}$ be the solenoid constructed this way, then the topology of the solenoid is just the subset topology induced by the Euclidean topology on ${\displaystyle \mathbb {R} ^{3}}$.

Since the parametrization is bijective, we can pullback the topology on ${\displaystyle S}$ to ${\displaystyle \mathbb {R} }$, which makes ${\displaystyle \mathbb {R} }$ itself the solenoid. This allows us to construct the inverse limit maps explicitly:

Construction by symbolic dynamics

Viewed as a set, the solenoid is just a Cantor-continuum of circles, wired together in a particular way. This suggests to us the construction by symbolic dynamics, where we start with a circle as a "racetrack", and append an "odometer" to keep track of which circle we are on.

Define ${\displaystyle S=S^{1}\times \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}}$ as the solenoid. Next, define addition on the odometer ${\displaystyle \mathbb {Z} \times \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}\to \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}}$, in the same way as p-adic numbers. Next, define addition on the solenoid ${\displaystyle +:\mathbb {R} \times S\to S}$ by

The topology on the solenoid is generated by the basis containing the subsets ${\displaystyle S'\times Z'_{(m_{1},...,m_{k})}}$, where ${\displaystyle S'}$ is any open interval in ${\displaystyle S^{1}}$, and ${\displaystyle Z'_{(m_{1},...,m_{k})}}$ is the set of all elements of ${\displaystyle \prod _{k=1}^{\infty }\mathbb {Z} _{n_{k}}}$ starting with the initial segment ${\displaystyle (m_{1},...,m_{k})}$.

Pathological properties

Solenoids are

the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space.

See also

• Protorus, a class of topological groups that includes the solenoids
• Pontryagin duality
• Inverse limit
• p-adic number
• Profinite integer

References

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (January 2012) (Learn how and when to remove this template message)

• D. van Dantzig, Ueber topologisch homogene Kontinua, Fund. Math. 15 (1930), pp. 102–125
• "Solenoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Clark Robinson, Dynamical systems: Stability, Symbolic Dynamics and Chaos, 2nd edition, CRC Press, 1998
  ISBN 978-0-8493-8495-0
• S. Smale, Differentiable dynamical systems, Bull. of the AMS, 73 (1967), 747 – 817.
• L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), pp. 454–472
• Robert F. Williams, Expanding attractors, Publ. Math. IHÉS, t. 43 (1974), p. 169–203

Further reading

• Bibcode:2012arXiv1201.2647S