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

TL;DR

This paper investigates the Hausdorff dimension of the set of points with unbounded orbits in cylindrical transformations, showing that for almost every rotation, this set can have dimension at least 1/2, and providing conditions for positive dimension.

Contribution

It establishes lower bounds and conditions for the Hausdorff dimension of unbounded orbit sets in cylindrical transformations, extending results to multidimensional rotations.

Findings

For almost every lpha, there exists i with Hausdorff dimension  at least 1/2.

A Diophantine condition on lpha guarantees positive dimension of the unbounded orbit set.

Constructed smooth i for multidimensional rotations with positive Hausdorff dimension.

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.