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

Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension