  For each irrational $\alpha\in[0,1)$ we construct a continuous function $f\:
[0,1)\to \R$ such that the corresponding cylindrical transformation
$[0,1)\times\R \ni (x,t) \mapsto (x+\alpha, t+ f(x)) \in [0,1)\times\R$ is
transitive and the Hausdorff dimension of the set of points whose orbits are
discrete is 2. Such cylindrical transformations are shown to display a certain
chaotic behaviour of Devaney-like type.
