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 $T_{\phi }:(x,t)\mapsto (x+\alpha ,t+\phi (x))$ where $Tx=x+\alpha$ 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 $\alpha$ there exists $\phi$ such that the Hausdorff dimension of $D(\alpha ,\phi )$ is at least $1/2$. We also provide a Diophantine condition on $\alpha$ that guarantees the existence of $\phi$ such that the dimension of $D(\alpha ,\phi )$ is positive. Finally, for some multidimensional rotations $T$ on $\T^d$, $d\ge 3$, we construct smooth $\phi$ so that the Hausdorff dimension of $D(\alpha ,\phi )$ is positive.

arXiv comment32 pages, 1 figure

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