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## Solenoids

A solenoid is a continuum homeomorphic to the inverse limit of the inverse sequence of unit circles in the complex plane with bonding maps , where is a sequence of prime numbers; it is called a -adic solenoid . The solenoid is known as a dyadic solenoid .

Geometrically, solenoid can be described as the intersection of a sequence of solid tori such that wraps times around without folding and is -thin, for each , where . See Figure A.

1. Each solenoid can be constructed (up to a homeomorphism) as the quotient space of the product by identifying each point with , where is a homeomorphism of the Cantor set such that for every there exist a closed-open subset of and a positive integer such that is a cover of consisting of pairwise disjoint subsets of with diameters less than [Gutek 1980].

2. Each solenoid is an Abelian topological group with a group operation and the neutral element .

3. Either of the following conditions is equivalent for a nondegenerate continuum different from a simple closed curve to be a solenoid.
1. is homeomorhic to a one-dimensional topological group [Hewitt 1963];

2. is indecomposable and is homeomorphic to a topological group [Chigogidze 1996, Theorem 8.6.18];

3. is circle-like, has the property of Kelley and contains no local end point [Krupski 1984c, Theorem (4.3)];
4. is circle-like, has the property of Kelley, each proper nondegenerate subcontinuum of is an arc and has no end pointsend point;

5. is circle-like, has the property of Kelley and has an open cover by Cantor bundles of open arcs (i.e., sets homeomorphic to the product of the Cantor set and the open interval ) [Krupski 1982];

6. is homogeneous, contains no proper, nondegenerate, terminal subcontinua and sufficiently small subcontinua of are not -ods [Krupski 1995, Theorem 3.1];

7. is a homogeneous curve containing an open subset such that some component of does not have the disjoint arcs property [Krupski 1995, p. 166];

8. is a homogeneous finitely cyclic (or, equivalently, -junctioned) curve that is not tree-like and contains no nondegenerate, proper, terminal subcontinua [Krupski et al. XXXXb], [Duda et al. 1991].

9. is openly homogeneous and sufficiently small subcontinua of are arcs [Prajs 1989];

4. Solenoid is a continuous image of if and only if the sequence is a factorant of sequence , i.e., there exists such that for each there is such that is a factor of .

Two solenoids are homeomorphic if and only if each of them is a continuous image of another [Cook 1967], [D. van Dantzig 1930, Satz 8, p. 122].

There is a family of solenoids of cardinality such that no member of the family is a continuous image of another.

5. Each monotonemonotone map image of a solenoid is homeomorphic to [Krupski 1984b, Theorem 5].

Each open map transforms onto a solenoid or onto an arc-like continuum with the property of Kelley and with arcs as proper nondegenerate subcontinua; if the map is a local homeomorphism, then its image is a solenoid [Krupski 1984a].

6. The composant of a solenoid containing is a one-parameter topological subgroup of , i.e. it is a one-to-one continuous homomorphic image of the additive group of the reals.

7. Any two composants of any two solenoids are homeomorphic [R. de Man 1995].

8. No solenoid can be mapped onto a strongly self-entwined continuum. In particular, it cannot be mapped onto a circle-like plane continuum which is a common part of a descending sequence of circular chains such that circles times in clockwisely and then times counter-clockwisely and the first link of contains the closure of the first link of [Rogers 1971b].

9. No movable continuum (in particular no continuum lying in a surface or a tree-like continuum) can be continuously mapped onto a solenoid. Alternatively, if the first Alexander-Cech cohomology group of a continuum is finitely divisiblefinitely divisible group, then cannot be mapped onto a solenoid [Krasinkiewicz 1976, Remark, p. 46, 4.1, 4.9., 5.1], [Krasinkiewicz 1978, Corollary 7.3], [Rogers 1975].

10. Every nonplanar, circle-like continuum has the shape of a solenoid [Krasinkiewicz 1976, remark, p. 46]. Two solenoids have the same shape if and only if they are homeomorphic [Godlewski 1970].

11. Any autohomeomorphism of is isotopic to a homeomorphism which is induced by a map of the inverse sequences which define ( can be a group translation, the involution, a power map or its inverse, or compositions of these maps). Maps and have equal the topological entropies and are semi-conjugate if the entropy is positive [Kwapisz 2001, Theorems 1-3, pp. 252-253], [D. van Dantzig 1930, Satz 9, p. 125].

The topological group of all autohomeomorphisms (with the compact-open topology) of a solenoid is homeomorphic (but not isomorphic) to the the product , where is the Hilbert space and the group of all topological group automorphisms of is equipped with the discrete topology and it is equal to , or , or [Keesling 1972, Theorems 3.1 and 2.4].

12. If the spaces of all autohomeomorphisms of two solenoids are homeomorphic, then the solenoids are isomorphic as topological groups [Keesling 1972, Corollary 3.9].

13. Any map is, for every , -homotopic to a map induced by a map between inverse sequences defining the corresponding solenoids [Rogers et al. 1971].

14. A -adic solenoid admits a mean if and only if infinitely many numbers in the sequence equal 2 [Krupski XXXXa]. The same condition is equivalent to the non-existence of exactly 2-to-1 map defined on the solenoid [Debski 1992].

15. The hyperspace of all subcontinua of any solenoid is homeomorphic the cone over [Rogers 1971a], [Nadler 1991, p. 202].

16. The family of all solenoids in the cube (as a subset of the hyperspace ) is Borel and not [Krupski XXXXc].

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Next: Compactifications of the real Up: Irreducible circle-like continua Previous: Irreducible circle-like continua
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30