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

Rigidity and Non-recurrence along Sequences

Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences

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

Analytic nonregular cocycles over irrational rotations

summary:Analytic cocycles of type $III_0$ over an irrational rotation are constructed and an example of that type is given, where all corresponding special flows are weakly mixing

Cohomology groups, multipliers and factors in ergodic theory

The problem of compact factors in ergodic theory and its relationship with the problem of extending a cocycle to a cocycle of a larger action are studied

Sur l'absence de mélange pour des flots spéciaux au-dessus d'une rotation irrationnelle

We prove the absence of mixing for special flows built over (1) an irrational rotation and under a function whose Fourier coefficients are of order O(1/|n|), and (2) an irrational rotation (satisfying a diophantine condition) and under a function having a finite number of singularities of a logarithmic type. These results generalize two theorems of Kochergin