On Hausdorff dimension of the set of closed orbits for a cylindrical transformation

Abstract

We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_φ:(x,t)\mapsto(x+α,t+φ(x))$ where $Tx=x+α$ is an irrational rotation on the circle $\T$ and $φ:\T\to\R$ is continuous, i.e.\ we try to estimate how big can be the set $D(α,φ):=\{x\in\T:|φ^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}$. We show that for almost every $α$ there exists $φ$ such that the Hausdorff dimension of $D(α,φ)$ is at least $1/2$. We also provide a Diophantine condition on $α$ that guarantees the existence of $φ$ such that the dimension of $D(α,φ)$ is positive. Finally, for some multidimensional rotations $T$ on $\T^d$, $d\geq3$, we construct smooth $φ$ so that the Hausdorff dimension of $D(α,φ)$ is positive.