arXiv subjects

K. Fraczek

Publications and source records attributed to K. Fraczek.

3 recordsLinked to original sources

arXiv2012

Non-reversibility and self-joinings of higher orders for ergodic flows

By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on the two torus, so called von Neumann's flows and some special flows with piecewise polynomial roof functions.. A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities.

math.DS↗

arXiv2010

Ratner's property and mixing for special flows over two-dimensional rotations

We consider special flows over two-dimensional rotations by $(α,β)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$ Such flows are shown to be always weakly mixing and never partially rigid. For an uncountable set of $(α,β)$ with both $α$ and $β$ of unbounded partial quotients the strong mixing property is proved to hold. It is also proved that while specifying to a subclass of roof functions and to ergodic rotations for which $α$ and $β$ are of bounded partial quotients the corresponding special flows enjoy so called weak Ratner's property. As a consequence, such flows turn out to be mildly mixing.

math.DS↗

arXiv2007

Mild mixing property for special flows under piecewise constant functions

We give a condition on a piecewise constant roof function and an irrational rotation by $α$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.

math.DS↗