{Solenoids}

A
\e{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
\es{$\mathbf p$-adic solenoid}{p-adic solenoid}
.  The solenoid $\Sigma(2,2,\dots)$ is known as a
\e{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{A} {dyadic selenoid}

\GIF{AA} {dyadic selenoid - an animation}

\GIF{AAA} {dyadic selenoid - an animation with a knot ;-)}

\rostere
\item
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
$\epsilon$ \cite{Gutek1980a}.

\item
Each solenoid $\Sigma(\mathbf p)$ is an Abelian
\g{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
$e=(1,1,\dots)$.

\item
Either of the following conditions is equivalent for a
nondegenerate continuum $X$ different from
a simple closed curve to be a solenoid.
\rostere

\item
$X$ is homeomorhic to a \g{one-dimensional}
topological group \cite{Hewitt+1963a};

\item
$X$ is  \g{indecomposable} and is homeomorphic to a topological group
\cite[Theorem 8.6.18]{Chigogidze1996a};

\item
 $X$ is  \g{circle-like}, has the
 \g{property of Kelley} and contains no
 \g{local end point} \cite[Theorem (4.3)]{Krupski1984c};
\item
$X$ is \g{circle-like}, has the \g{property of Kelley}, each proper
nondegenerate subcontinuum of $X$ is an arc and $X$ has no
\g{end points}{end point};

\item
$X$ 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 $(0,1)$) \cite{Krupski1982a};

\item
$X$ is \g{homogeneous}, contains no proper,
nondegenerate, \gs{terminal}{terminal subcontinuum} subcontinua
and sufficiently small subcontinua of $X$ are not
\gs{$\infty$-ods}{infty-od} \cite[Theorem
3.1]{Krupski1995a};

\item
$X$ is a homogeneous curve containing an open subset $U$
such that  some component of $U$ does not have the
\g{disjoint arcs property} \cite[p.
166]{Krupski1995a};

\item
$X$ is a homogeneous \gs{finitely cyclic}{finitely cyclic
curve}  (or, equivalently, \gs{$k$-junctioned}{k-junctioned
curve}) curve  that is not tree-like and contains no
nondegenerate, proper, terminal subcontinua
\cite{Krupski+XXXXb}, \cite{Duda+1991a}.

\item
$X$ is \g{openly homogeneous} and
sufficiently small subcontinua of $X$ are arcs
\cite{Prajs1989a};


\rosteree




\item
 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
 \g{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
\cite{Cook1967a}, \cite[Satz 8, p. 122]{Dantzig1930a}.

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

\item
Each monotone\g{monotone map} image of a solenoid $X$ is
homeomorphic to $X$ \cite[Theorem 5]{Krupski1984b}.

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 \cite{Krupski1984a}.




\item
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.


\item
Any two \gs{composants}{composant} of any two solenoids are
homeomorphic \cite{Man1995a}.

%\item
%Composants of solenoids are not planable.

\item
No solenoid can be mapped onto a \g{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 $C_{i+1}$ \cite{Rogers1971b}.

\item
No \gs{movable}{movable continuum} continuum (in particular no
continuum lying in a surface or a \gs{tree-like}{tree-like
continuum} continuum) can be continuously mapped onto a
solenoid. Alternatively, if the  first Alexander-\v Cech
cohomology group of a  continuum $X$ is finitely
divisible\g{finitely divisible group}, then $X$ cannot be
mapped onto a solenoid \cite[Remark, p. 46, 4.1, 4.9.,
5.1]{Krasinkiewicz1976a}, \cite[Corollary
7.3]{Krasinkiewicz1978a}, \cite{Rogers1975a}.

\item
Every nonplanar, circle-like continuum has the
\g{shape} of a solenoid \cite[remark, p.
46]{Krasinkiewicz1976a}. Two solenoids  have the same shape
if and only if they are homeomorphic
  \cite{Godlewski1970a}.

 \item
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 \gs{topological
entropies}{topological entropy} and are
\g{semi-conjugate} if the entropy is positive
\cite[Theorems 1--3, pp. 252--253]{Kwapisz2001a}, \cite[Satz
9, p. 125]{Dantzig1930a}.

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 $\mathbb Z_2\oplus_{i=1}^\infty\mathbb Z$
 \cite[Theorems 3.1 and 2.4]{Keesling1972a}.

\item
If the spaces of all autohomeomorphisms of two solenoids are homeomorphic,
then the solenoids are isomorphic as topological groups
\cite[Corollary 3.9]{Keesling1972a}.

\item
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 \cite{Rogers+1971a}.




\item
A $\mathbf p$-adic solenoid admits a \g{mean} if and
only if infinitely many numbers in the sequence $\mathbf p$
equal 2 \cite{KrupskiXXXXa}.  The same condition is
equivalent to the non-existence of exactly 2-to-1 map
defined on the solenoid \cite{Debski1992a}.


\item
The \gs{hyperspace}{hyperspace of subcontinua} of all subcontinua of any
solenoid $\Sigma$ is homeomorphic  the cone over $\Sigma$
\cite{Rogers1971a}, \cite[p. 202]{Nadler1991a}.

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

\rosteree
