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Solenoid (mathematics)
Encyclopedia
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 group
s and continuous
homomorphism
s
, fi: Si+1 → Si, i ≥ 0,
where each Si is a circle
and fi is the map that uniformly wraps the circle Si+1 ni times (ni ≥ 2) around the circle Si. 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 ni 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 system
s.
Fix a sequence of natural numbers {ni}, 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 {ni}.
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
This map is a smooth embedding
of T into itself that preserves the foliation
by meridional disks (the constants 1/2 and 1/4 are somewhat arbitrary, but it is essential that 1/4 < 1/2 and 1/4 + 1/2 < 1). 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 set
Λ of the discrete dynamical system (T, f) 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 (in the sense of topological dimension) attractor
, and the dynamics of f on Λ has the following interesting properties:
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.
metrizable spaces that are connected
, but not locally connected or path connected. This is reflected in their pathological
behavior with respect to various homology theories, in contrast with the standard properties of homology for simplicial complex
es. In Čech homology
, one can construct a non-exact long homology sequence using a solenoid. In Steenrod-style homology theories, the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a solenoid is a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(i.e. a continuum
Continuum (topology)
In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...
) that may be obtained as the inverse limit
Inverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
of an inverse system of topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s and continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s
, fi: Si+1 → Si, i ≥ 0,
where each Si is a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
and fi is the map that uniformly wraps the circle Si+1 ni times (ni ≥ 2) around the circle Si. This construction can be carried out geometrically in the three-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
R3. A solenoid is a one-dimensional homogeneous indecomposable continuum
Indecomposable continuum
In point-set topology, an indecomposable continuum is a continuum that is not the union of any two of its proper subcontinua. The pseudo-arc is an example of a hereditarily indecomposable continuum. L. E. J...
that has the structure of a compact topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
.
In the special case where all ni 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
Leopold Vietoris
Leopold Vietoris was an Austrian mathematician and a World War I veteran who gained additional fame by becoming a supercentenarian...
for n = 2 and by van Dantzig
David van Dantzig
David van Dantzig was a Dutch mathematician, well known for the construction in topology of the dyadic solenoid....
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 system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s.
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 {ni}, ni ≥ 2. Let T0 = S1 × D be a solid torus
Solid torus
In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
. 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 {ni}.
Here is a variant of this construction isolated by Stephen Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
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
This map is a smooth embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
of T into itself that preserves the foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
by meridional disks (the constants 1/2 and 1/4 are somewhat arbitrary, but it is essential that 1/4 < 1/2 and 1/4 + 1/2 < 1). 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 set
Hyperbolic set
In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under f, with respect to some...
Λ of the discrete dynamical system (T, f) 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 (in the sense of topological dimension) attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
, and the dynamics of f on Λ has the following interesting properties:
- meridional disks are the stable manifoldStable manifoldIn mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor...
s, each of which intersects Λ over a Cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.... - periodic pointPeriodic pointIn mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.- Iterated functions :...
s of f are 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
Branched manifold
In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restriced type and admits a well-defined tangent space at each point...
in place of the circle, together with an expanding self-immersion.
Pathological properties
Solenoids are compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
metrizable spaces that are connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, but not locally connected or path connected. This is reflected in their pathological
Pathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved....
behavior with respect to various homology theories, in contrast with the standard properties of homology for simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es. In Čech homology
Cech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.-Motivation:...
, one can construct a non-exact long homology sequence using a solenoid. In Steenrod-style homology theories, the 0th homology group of a solenoid may have a fairly complicated structure, even though a solenoid is a connected space.