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\documentclass{era-l}
\title{ On composants of solenoids }
%\footnotetext{1991 {\it Mathematics Subject Classification:} Primary 54F15;
%Secondary 54F65, 54H20.}
%}
\author{Ronald de Man}
\subjclass{Primary 54F15; Secondary 54F65, 54H20}
\communicated_by{Krystyna Kuperberg}
\date{June 26, 1995}
\address{Faculteit Wiskunde en Informatica, 
TU Eindhoven,
Postbus 513,
5600 MB Eindhoven,
The Netherlands}
\email{deman@win.tue.nl} 
\def\currentvolume{1}
\def\currentissue{2}

\begin{document}

\setcounter{page}{87}

\makeatletter
\def\eqalign#1{\null\,\vcenter{\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\makeatother

\let\epsilon=\varepsilon

\newcommand{\N}{{\bf N}}
\newcommand{\Z}{{\bf Z}}
\newcommand{\R}{{\bf R}}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}

%\newenvironment{proof}{{\it Proof.}}{\hfill$\Box$}

%\maketitle

\begin{abstract}
\noindent
It is proved that any two composants of any two solenoids are
homeomorphic.
\end{abstract}

\maketitle

\section{Introduction}
Solenoids were introduced by Van Dantzig \cite{DD}. His original
description runs as follows. Let $P=(p_1,p_2,\ldots)$ be a sequence
of primes. The solenoid $S_P$ is the intersection of a descending
sequence of solid tori $T_1\supset T_2\supset T_3\supset \ldots$
such that $T_{i+1}$ is wrapped around inside $T_i$ longitudinally
$p_i$ times without folding back. Van Heemert proved that solenoids are
indecomposable continua \cite{AH}.

For solenoids a classification theorem exists, conjectured by Bing \cite{RB}
and proved by McCord \cite{McC}, giving necessary and sufficient conditions
for two solenoids $S_P$ and $S_Q$ to be homeomorphic (see also \cite{A-F}).

The composants of solenoids coincide with the arc components. Since
solenoids are topological groups, any two composants of the same
solenoid are homeomorphic. The main theorem of this paper is
\begin{theorem}
Any two composants of any two solenoids are homeomorphic.
\end{theorem}

The composants of solenoids are examples of orbits in dynamical systems
that are not locally compact. A locally compact orbit is either a
singleton, a simple closed curve, or a topological copy of the
real line. For orbits which are not locally compact the situation is
much more complicated. Fokkink has proved the existence of uncountably
many of them \cite{RF1,RF2}.

In \cite{CB} Bandt shows that any two nonzero composants of the bucket handle
are homeomorphic. Following a suggestion of Fokkink, we shall adapt the
ideas from that article to prove the result of this paper.

For the proof, we need a different description of solenoids. We define the
cascade $(C_P,\sigma)$ as follows. $C_P$ is the Cantor set represented
as the topological product $C_P=\prod_{i=1}^\infty{\overline p_i}$ of
discrete spaces ${\overline p_i}=\{0,1,\ldots,p_i-1\}$. The homeomorphism
$\sigma:C_P\to C_P$ has the form
$$
\eqalign{
&\sigma(x_1,x_2,\ldots)=(x_1+1,x_2,x_3,\ldots)\quad\mbox{if $x_1x\}$.
If there exist clopen subsets $A$ and $B$ of $I$ and $J$, respectively,
and a homeomorphism $f:A\to B$ such that $f\circ\tau_A=\tau_B\circ f$,
we say that $(I,\tau)$ and $(J,\tau)$ are {\it first return equivalent}.
>From Theorem 5.2 in \cite{A} it follows that $(I,\tau)$ and $(J,\tau)$
are first return equivalent if and only if $\Sigma(I,\tau)$ and
$\Sigma(J,\tau)$ are homeomorphic.

Theorem 1 is proved by constructing clopen subsets $A$ and $B$. In the
following section we sketch this construction. The details will
appear elsewhere \cite{AJM}.

\section{Sketch of the construction}
First we choose two sequences of integers $0=n_0