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

For each irrational $\alpha \in [0,1)$ we construct a continuous function $f: [0,1)\to \R$suchthatthecorrespondingcylindricaltransformation$[0,1)\times\R \ni (x,t) \mapsto (x+\alpha, t+ f(x)) \in [0,1)\times\R$ 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.