Mariusz Lemańczyk

Absolutely continuous cocycles over irrational rotations

AbstractFor homeomorphisms

Ergodicity of Rokhlin cocycles

\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)

Transformations conjugate to their inverses have even essential values

(z, w ∈S1,α is irrational,ϕ:S1→S1) of the torusS1×S1 it is proved thatTϕ has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions wheneverϕ is absolutely continuous with nonzero topological degree and the derivative ofϕ is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.

On mild mixing of special flows over irrational rotations under piecewise smooth functions

We consider methods of establishing ergodicity of group extensions, proving that a class of cylinder flows are ergodic, coalescent and non-squashable. A new Koksma-type inequality is also obtained.

Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces

We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such groups. Finally, we examine the structure of self-joinings of extensions.We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss on the class of automorphisms disjoint from all weakly mixing transformations.Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker dynamical systems.

On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

Assuming that two ergodic automorphisms T and R are disjoint, we prove that an ergodic extension

A class of special flows over irrational rotations which is disjoint from mixing flows

\tilde{T}

A note on the existence of a largest topological factor with zero entropy

of T remains disjoint with R when, in a representation of