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

Abstract

For each irrational $\alpha \in [0,1)$ we construct a continuous function $f[0,1)\to \mathbb{R}$ such that the corresponding cylindrical transformation $[0,1)\times \mathbb{R}\ni (x,t)\mapsto (x+\alpha ,t+f(x))\in [0,1)\times \mathbb{R}$ 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.

14 pages