Solenoid (math)

A Smale-Williams sol

In mathematics , solenoids are certain continua that appear, among other things, as attractors in the theory of dynamic systems .

definition

A solenoid is a topological group that represents the projective limit of a sequence of continuous homomorphisms

${\displaystyle f_{ij}\colon S_{i}\to S_{j}}$

where all topological groups are homeomorphic to the circle group . ${\displaystyle S_{i}}$

If one realizes the circle group as , then all are of the form ${\displaystyle \left\{z\in \mathbb {C} \colon |z|=1\right\}}$ ${\displaystyle f_{ij}}$

${\displaystyle f_{ij}(z)=z^{n_{ij}}}$

for a . To put it clearly, the circle wraps around itself, depending on the sign of in a positive or negative direction. ${\displaystyle n_{ij}\in \mathbb {Z} }$ ${\displaystyle f_{ij}}$ ${\displaystyle \mid n_{ij}\mid }$ ${\displaystyle n_{ij}}$

properties

Solenoids are compact , contiguous, and one-dimensional . They are indivisible continua and not locally or locally connected . They can be embedded in three-dimensional Euclidean space and can therefore be metrised .

Examples

First step in the construction of a Smale-Williams solenoid: a full torus is wrapped around twice within the full torus . ${\displaystyle T_{2}}$ ${\displaystyle T_{1}}$

The First Six Steps in the Construction of a Smale-Williams Solenoid.

• The following topological groups are all isomorphic to each other and are a solenoid:

1. the projective limit , where is partially ordered by the divisibility relation and is given for the mapping from the -th to the -th copy of through ; ${\displaystyle \varprojlim _{n\in \mathbb {N} }S^{1}}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle m\mid n}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle S^{1}}$ ${\displaystyle z\to z^{\frac {n}{m}}}$
2. the projective limit , which is partially ordered by the divisibility relation and is induced for the mapping from to by the identity mapping ; ${\displaystyle \varprojlim _{n\in \mathbb {N} }\mathbb {R} /{\frac {1}{n}}\mathbb {Z} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle m\mid n}$ ${\displaystyle \mathbb {R} /{\frac {1}{n}}\mathbb {Z} }$ ${\displaystyle \mathbb {R} /{\frac {1}{m}}\mathbb {Z} }$
3. the Pontryagin dual , d. H. the set of group homomorphisms with the compact-open topology , with the discrete topology bearing; ${\displaystyle \mathbb {Q} ^{\vee }}$ ${\displaystyle \mathbb {Q} \to S^{1}}$ ${\displaystyle \mathbb {Q} }$
4. the Adele class group , with the Adelring and embedded diagonally. ${\displaystyle {\mathbb {A} }/\mathbb {Q} }$ ${\displaystyle {\mathbb {A} }}$ ${\displaystyle \mathbb {Q} \subset {\mathbb {A} }}$

• The Smale-Willians solenoid for a sequence of natural numbers is constructed as follows: start with a full torus , then a full torus is wound around within times (the picture on the right shows the case ), then a full torus is wound around within times, and immediately. The diameter of the cross-section of the full gate should converge to zero. The intersection is then homeomorphic to the solenoid defined by the sequence . ${\displaystyle n_{i}}$ ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$ ${\displaystyle T_{1}}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{1}=2}$ ${\displaystyle T_{3}}$ ${\displaystyle T_{2}}$ ${\displaystyle n_{2}}$ ${\displaystyle \Lambda =\bigcap T_{i}}$ ${\displaystyle f_{i,i+1}(z)=z^{n_{i}}}$

literature

• Leopold Vietoris : About the higher context of compact spaces and a class of contextual images. Math. Ann. 97: 454-472 (1927).
• David van Dantzig : About topologically homogeneous continua. Find. Math. 15: 102-125 (1930).

Web links

• Solenoid (Encyclopedia of Mathematics)

Individual evidence

1. ↑ Robert Kucharczyk, Peter Scholze : Topological realizations of absolute Galois groups online
2. ^ Robert F. Williams: Expanding attractors. Publ. Math. IHES 43 (1974), 169-203