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A solenoid is a continuum homeomorphic to the inverse limit \Sigma(\mathbf p)=\varprojlim(S^1,f_n)$ of the inverse sequence of unit circles S^1$ in the complex plane with bonding maps f_n(z)=z^{p_n}$, where \mathbf
p=(p_1,p_2,\dots)$ is a sequence of prime numbers; it is called a \mathbf p$-adic solenoid . The solenoid \Sigma(2,2,\dots)$ is known as a dyadic solenoid .

Geometrically, solenoid \Sigma(\mathbf p)$ can be described as the intersection of a sequence of solid tori T_1\supset T_2\supset\dots$ such that T_{n+1}$ wraps p_n$ times around T_n$ without folding and T_n$ is \epsilon_n$-thin, for each n \in \mathbb{N}$, where \lim\epsilon_n=0$. See Figure A.

Figure 4.3.1: ( A ) dyadic selenoid

Figure: ( AA ) dyadic selenoid - an animation

Figure: ( AAA ) dyadic selenoid - an animation with a knot ;-)

  1. Each solenoid can be constructed (up to a homeomorphism) as the quotient space of the product C\times I$ by identifying each point (c,1)$ with (h(c),0)$, where h:C\to C$ is a homeomorphism of the Cantor set C$ such that for every \epsilon>0$ there exist a closed-open subset D$ of C$ and a positive integer n$ such that \{f\sp
j(D)\colon j=1,\cdots,n\}$ is a cover of C$ consisting of pairwise disjoint subsets of C$ with diameters less than \epsilon$ [Gutek 1980].

  2. Each solenoid \Sigma(\mathbf p)$ is an Abelian topological group with a group operation (z_1,z_2,\dots)\cdot (z'_1,z'_2,\dots)=(z_1\cdot
z'_1,z_2\cdot z'_2,\dots)$ and the neutral element e=(1,1,\dots)$.

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

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

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

    5. X$ 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 C\times(0,1)$ of the Cantor set C$ and the open interval (0,1)$) [Krupski 1982];

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

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

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

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

  4. Solenoid \Sigma(\mathbf q)$ is a continuous image of \Sigma(\mathbf p)$ if and only if the sequence \mathbf q=(q_1,q_2,\dots)$ is a factorant of sequence \mathbf
p=(p_1,p_2,\dots)$, i.e., there exists i$ such that for each j\ge i$ there is k$ such that q_i\cdot \dots\cdot q_j$ is a factor of p_1\cdot\dots\cdot p_k$.

    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 2^{\aleph_0}$ such that no member of the family is a continuous image of another.

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

    Each open map transforms X$ 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 \Sigma$ containing e$ is a one-parameter topological subgroup of \Sigma$, 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 C_i$ such that C_{i+1}$ circles n$ times in C_i$ clockwisely and then n-1$ times counter-clockwisely and the first link of C_i$ contains the closure of the first link of C_{i+1}$ [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 X$ is finitely divisiblefinitely divisible group, then X$ 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 f$ of \Sigma(\mathbf p)$ is isotopic to a homeomorphism g$ which is induced by a map (S^1,f_n)\to (S^1,f_n)$ of the inverse sequences which define \Sigma(\mathbf p)$ (g$ can be a group translation, the involution, a power map or its inverse, or compositions of these maps). Maps f$ and g$ 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 \Sigma$ is homeomorphic (but not isomorphic) to the the product \Sigma\times l_2\times Aut(\Sigma)$, where l_2$ is the Hilbert space and the group Aut(\Sigma)$ of all topological group automorphisms of \Sigma$ is equipped with the discrete topology and it is equal to \mathbb{Z}_2$, or \mathbb{Z}_2\times\mathbb{Z}^n$, or \mathbb{Z}_2\oplus_{i=1}^\infty\mathbb{Z}$ [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 f:\Sigma(\mathbf p)\to \Sigma(\mathbf q)$ is, for every \epsilon>0$, \epsilon$-homotopic to a map induced by a map (S^1,f_n)\to (S^1,g_n)$ between inverse sequences defining the corresponding solenoids [Rogers et al. 1971].

  14. A \mathbf p$-adic solenoid admits a mean if and only if infinitely many numbers in the sequence \mathbf p$ 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 \Sigma$ is homeomorphic the cone over \Sigma$ [Rogers 1971a], [Nadler 1991, p. 202].

  16. The family of all solenoids in the cube I^3$ (as a subset of the hyperspace C(I^3)$) is Borel and not G_{\delta\sigma}$ [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