New examples of Besicovitch transitive cylindrical cascades

New examples of transitive cylindrical cascades with discrete orbits (the Besicovitch property) are constructed. For each $\gamma\in(0,1)$ there exists a cylindrical cascade over a rotation of the circle, with a $\gamma$-Hölder continuous function, that has the Besicovitch property; furthermore, the Hausdorff dimension of the set of points on the circle which have discrete orbits is at least $1-\gamma$. This improves (by $\varepsilon$) an earlier estimate. In addition, an example of a cascade with discrete orbits such that the corresponding function satisfies the Hölder condition with each exponent $\gamma\in(0,1)$ is constructed.
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