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

Mathematics – Dynamical Systems

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

32 pages, 1 figure

Scientific paper

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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

On Hausdorff dimension of the set of closed orbits for a cylindrical transformation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On Hausdorff dimension of the set of closed orbits for a cylindrical transformation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Hausdorff dimension of the set of closed orbits for a cylindrical transformation will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-277890

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.