Solenoid

Let $ \mathbf n = \langle n _ {i} \rangle _ {i} $ be a sequence of positive integers. From $ \mathbf n $ one constructs a topological space as follows.

Let $ T _ {0} $ be a torus in $ \mathbf R ^ {3} $; inside $ T _ {0} $ one takes a torus $ T _ {1} $ wrapped around longitudinally $ n _ {1} $ times, in a smooth fashion without folding back; inside $ T _ {1} $ one takes a torus $ T _ {2} $ wrapped around $ n _ {2} $ times in the same way. Continuing this procedure indefinitely, one obtains a decreasing sequence of tori. Its intersection is called the $ \mathbf n $- adic solenoid $ \Sigma _ {\mathbf n } $.

The basic properties of $ \Sigma _ {\mathbf n } $ are that it is a one-dimensional continuum which, moreover, is indecomposable (cf. Indecomposable continuum).

$ \Sigma _ {\mathbf n } $ is also a topological group; this can be seen if one considers an alternative construction of $ \Sigma _ {\mathbf n } $ as the inverse limit of the following inverse sequence:

$$ {} \dots S _ {3} \rightarrow ^ { {f _ 3} } S _ {2} \mathop \rightarrow \limits ^ { {f _ {2} }} S _ {1} \rightarrow ^ { {f _ 1} } S _ {0} , $$

where each $ S _ {i} $ is the unit circle and $ f _ {i} : S _ {i} \rightarrow S _ {i-} 1 $ is defined by $ f _ {i} ( z)= z ^ {n _ {i} } $. There are various other ways in which one can construct the solenoids, see, e.g., [a3].

Solenoids were first defined by L. Vietoris [a2] (for the sequence $ \langle 2, 2 ,\dots \rangle $) and by D. van Dantzig [a1] (for all constant sequences).

Solenoids are also important in topological dynamics; on them one can define a flow (continuous-time dynamical system) structure [a4] which has a locally disconnected minimal set of almost-periodic motions.

There is a complete classification of the solenoids: first, without loss of generality one may assume that the numbers $ n _ {i} $ are prime. Call two sequences of primes $ \mathbf p $ and $ \mathbf q $ equivalent if one can delete from each a finite number of terms such that in the reduced sequences $ \mathbf p ^ \prime $ and $ \mathbf q ^ \prime $ every prime is counted the same number of times. One can then show that $ \Sigma _ {\mathbf p } $ and $ \Sigma _ {\mathbf q } $ are homeomorphic if and only if $ \mathbf p $ and $ \mathbf q $ are equivalent. See [a5] and [a6].

Finally one can characterize the solenoids as those metric continua that are homogeneous and have the property that every proper subcontinuum is an arc. See [a7].

References

[a1]	D. van Dantzig, "Ueber topologisch homogene Kontinua" Fund. Math. , 15 (1930) pp. 102–125
[a2]	L. Vietoris, "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen" Math. Ann. , 97 (1927) pp. 454–472
[a3]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1979)
[a4]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[a5]	R.H. Bing, "A simple closed curve is the only homogeneous bounded plane continuum that contains an arc" Canad. Math. J. , 12 (1960) pp. 209–230
[a6]	M.C. McCord, "Inverse limit sequences with covering maps" Trans. Amer. Math. Soc. , 114 (1965) pp. 197–209
[a7]	C.L. Hagopian, "A characterization of solenoids" Pacific J. Math. , 68 (1977) pp. 425–435