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| Let $X$ be a compact metric space and $T:X\longrightarrow X$ be a
homeomorphism of $X$. Let $\varphi:X\longrightarrow\mathbb{R}$ be
a continuous function (called \emph{a cocycle}). By a
\emph{cylinder transformation} we mean a homeomorphism
$T_\varphi:X\times\mathbb{R}\longrightarrow X\times\mathbb{R}$ (or
rather a $\mathbb{Z}$--action generated by it) given by the
formula
$$T_\varphi(x,r)=(Tx,\varphi(x)+r).$$
Such a transformation cannot be itself minimal (Besicovitch 1951, Le Calvez and Yoccoz 1997). If a base transformation is a rotation on a compact metric group then $T_\varphi$ is topologically ergodic iff $\varphi$ has zero mean with respect to the Haar measure and is not a coboundary. Fr\c{a}czek and Lema\'{n}czyk (2010) asked whether there exist topologically ergodic transformations having either dense or discrete orbits. We are able to construct such an example with $(X,T)$ being an odometer. |
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