Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincar\'e sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.Comment: 45 pages, 2 figure

Remarks on step cocycles over rotations, centralizers and coboundaries

By using a cocycle generated by the step function $\varphi_{\beta, \gamma} = 1_{[0, \beta]} - 1_{[0, \beta]} (. + \gamma)$ over an irrational rotation $x \to x + \alpha \mod 1$, we present examples which illustrate different aspects of the general theory of cylinder maps. In particular, we construct non ergodic cocycles with ergodic compact quotients, cocycles generating an extension $T_{\alpha, \varphi}$ with a small centralizer. The constructions are related to diophantine properties of $\alpha, \beta, \gamma$

Centralizer and liftable centralizer of special flows over rotations

The liftable centralizer for special flows over irrational rotations is studied. It is shown that there are such flows under piecewise constant roof functions which are rigid and whose liftable centralizer is trivial

On multiple ergodicity of affine cocycles over irrational rotations

Let T_\alpha denote the rotation T_{\alpha}x=x+\alpha (mod 1) by an irrational number \alpha on the additive circle T=[0,1). Let \beta_1,..., \beta_d be d\geqslant 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in R^(d+1)) generated over T_\alpha by the vectorial function \Psi_{d+1}(x):=(\phi(x), \phi(x+\beta_1),..., \phi(x+\beta_d)), with \phi(x)={x}-1/2. It was already proved in \cite{LeMeNa03} that \Psi_{2} is regular for \alpha with bounded partial quotients. In the present paper we show that \Psi_{2} is regular for any irrational \alpha. For higher dimensions, we give sufficient conditions for regularity. While the case d=2 remains unsolved, for d=3 we provide examples of non-regular cocycles \Psi_{4} for certain values of the parameters \beta_1,\beta_2,\beta_3. We also show that the problem of regularity for the cocycle \Psi_{d+1} reduces to the regularity of the cocycles of the form \Phi_{d} =(1_{[0, \beta_j]} - \beta_j)_{j= 1, ..., d} (taking values in R^d). Therefore, a large part of the paper is devoted to the classification problems of step functions with values in R^{d}.Comment: 34 pages; revisions in response to referees' comment

Limit law for some modified ergodic sums

An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems

Drilling Information System (DIS) and Core Scanner

The Drilling Information System is a modular structure of databases, tailored user applications as well as web services and instruments including appropriate interfaces to DIS. This tool set has been developed for geoscientific drilling projects but is applicable to other distributed scientific operations. The main focuses are the data acquisition on drill sites (ExpeditionDIS), and the curation of sample material e.g., in core repositories (CurationDIS). Due to the heterogeneity of scientific drilling projects, a project-specific DIS is arranged and adjusted from a collection of existing templates and modules according to the user requirements during a one week training course. The collected data are provided to the Science Team of the drilling project by secured Web services, and stored in long-term archives hosted at GFZ. At the end the data sets and sample material are documented in an Operational Report (e.g., Lorenz et al., 2015) and published with assigned DOI (Digital Object Identifier) and IGSN (International Geo Sample Number; for physical samples) by GFZ Data Services

Ergodicity for Infinite Periodic Translation Surfaces

For a Z-cover of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic