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

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.Comment: 32 pages, 1 figur

A TaqMan real-time PCR assay for Rhizoctonia cerealis and its use in wheat and soil

Rhizoctonia cerealis causes sharp eyespot in cereals and the pathogen survives as mycelia or sclerotia in soil. Real-time Polymerase Chain Reaction (qPCR) assays based on TaqMan chemistry are highly suitable for use on DNA extracted from soil. We report here the first qPCR assay for R. cerealis using TaqMan primers and a probe based on a unique Sequence Characterised Amplified Region (SCAR). The assay is highly specific and did not amplify DNA from a range of other binucleate Rhizoctonia species or isolates of anastomosis groups of Rhizoctonia solani. The high sensitivity of the assay was demonstrated in soils using a bulk DNA extraction method where 200 μg sclerotia in 50 g of soil were detected. DNA of the pathogen could also be amplified from asymptomatic wheat plants. Using the assay on soil samples from fields under different crop rotations, R. cerealis was most frequently detected in soils where wheat was grown or soil under pasture. It was detected least frequently in fields where potatoes were grown. This study demonstrates that assays derived from SCAR sequences can produce specific and sensitive qPCR assays