| Abstract: |
| Let $X$ be a compact metric monothetic group and $T:X\longrightarrow X$ be a
homeomorphism of $X$. Let $f:X\longrightarrow\mathbb{R}$ be
a continuous function (called \emph{a cocycle}). By a
\emph{cylinder transformation} we mean a homeomorphism.
$T_f:X\times\mathbb{R}\longrightarrow X\times\mathbb{R}$ (or
rather a $\mathbb{Z}$--action generated by it) given by the
formula
$$T_f(x,r)=(Tx,f(x)+r).$$
In \cite{P} H. Poincar\`{e} addressed the problem of what types of orbits may coexist in such a system.
We give a very brief history of the problem and describe some recent results.
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\small
\bibitem{P} H. Poincar\`{e}, \emph{M\`{e}moire sur les courbes d\`{e}finies par une \`{e}quation diff\`{e}rentielle},
1882.
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