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 $T_\varphi:(x,t)\mapsto(x+\alpha,t+\varphi(x))$ where $Tx=x+\alpha$ is an irrational rotation on the circle $\T$ and $\varphi:\T\to\R$ is continuous, i.e.\ we try to estimate how big can be the set $D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}$. We show that for almost every $\alpha$ there exists $\varphi$ such that the Hausdorff dimension of $D(\alpha,\varphi)$ is at least $1/2$. We also provide a Diophantine condition on $\alpha$ that guarantees the existence of $\varphi$ such that the dimension of $D(\alpha,\varphi)$ is positive. Finally, for some multidimensional rotations $T$ on $\T^d$, $d\geq3$, we construct smooth $\varphi$ so that the Hausdorff dimension of $D(\alpha,\varphi)$ is positive.