Besicovitch Cylindrical Transformation with a Hölder Function

For any $\gamma\in(0,1)$ and $\varepsilon>0$, we construct a cylindrical cascade with a $\gamma$-Hölder function over some rotation of the circle. This transformation has the Besicovitch property; i.e., it is topologically transitive and has discrete orbits. The Hausdorff dimension of the set of points of the circle that have discrete orbits is greater than $1-\gamma-\varepsilon$.