Krzysztof Frączek

On the degree of cocycles with values in the groupSU(2)

AbstractIn this paper are presented some properties of smooth cocycles over irrational rotations on the circle with values in the groupSU(2). It is proved that the degree of anyC2-cocycle (the notion of degree was introduced in [2]) belongs to 2πℕ (ℕ={0, 1, 2,...}). It is also shown that if the rotation satisfies a Diophantine condition, then everyC∞-cocycle with nonzero degree isC∞-cohomologous to a cocycle of the form

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

\mathbb{T} \ni x \mapsto \left[ \begin{gathered} e^{2\pi i(rx + w)} 0 \hfill \\ 0 e^{2\pi i(rx + w)} \hfill \\ \end{gathered} \right] \in SU(2)

Disjointness of interval exchange transformations from systems of probabilistic origin

, where 2πr is the degree of the cocycle andw is a real number. The above statement is false in the case of cocycles with zero degree. The proofs are based on ideas presented by R. Krikorian in [6].

On ergodicity of foliations on

We show that the vertical light rays in almost every periodic array of Eaton lenses do not leave certain strips of bounded width. The light rays are traced by leaves of a non-orientable foliation on a singular plane. We study the flow defined by the induced foliation on the orientation cover of the singular plane. The behaviour of that flow and ultimately our claim for the light rays are based on an analysis of the Teichmuller flow and the Kontsevich–Zorich cocycle on the moduli space of two-branched, two-sheeted torus covers in genus 2.

\mathbb Z^d

We deal with Besicovitchs problem of the existence of discrete orbits for transitive cylindrical transformations T : (x, t) (x + α, t + (x)), where Tx = x+α is an irrational rotation on the circle and is continuous, i.e. we try to estimate how big the set can be. We show that for almost every α there exists such that the Hausdorff dimension of D(α, ) is at least 1/2. We also provide a Diophantine condition on α that guarantees the existence of such that the dimension of D(α, ) is positive. Finally, for some multidimensional rotations T on , d ≥ 3, we construct smooth so that the Hausdorff dimension of D(α, ) is positive.

-covers of half-translation surfaces and some applications to periodic systems of Eaton lenses

Answering a question of A. Vershik we construct two non-weakly isomorphic ergodic automorphisms for which the associated unitary (Koopman) representations areMarkov quasi-similar.We also discuss metric invariants of Markov quasi-similarity in the class of ergodic automorphisms.

On special flows over IETs that are not isomorphic to their inverses

It is proved that almost every interval exchange transformation given by the symmetric permutation 1->m, 2->m-1,..., m-1->2, m->1, where m>1 is an odd number, is disjoint from ELF systems. The notion of ELF systems was introduced to express the fact that a given system is of probabilistic origin; the following standard classes of systems of probabilistic origin enjoy the ELF property: mixing systems, ergodic Gaussian systems, Poisson suspensions, dynamical systems coming from stationary infinitely divisible processes. Some disjointness properties of special flows built over interval exchange transformations and under piecewise constant roof function are investigated as well.

Non-ergodic

We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for