JMD
Ratner's property and mild mixing for special flows over two-dimensional rotations
Krzysztof Frączek Mariusz Lemańczyk
Journal of Modern Dynamics 2010, 4(4): 609-635 doi: 10.3934/jmd.2010.4.609
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)dxdy\ne 0$ or $\int_{\T^2}f_y(x,y)dxdy\ne 0 $. Such flows are shown to be always weakly mixing and never partially rigid. It is proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy the so-called weak Ratner property. As a consequence, such flows turn out to be mildly mixing.
keywords: Ratner's property. Measure-preserving flows mild mixing special flows
DCDS
The Chowla and the Sarnak conjectures from ergodic theory point of view
El Houcein El Abdalaoui Joanna Kułaga-Przymus Mariusz Lemańczyk Thierry de la Rue
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 2899-2944 doi: 10.3934/dcds.2017125

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in $\{-1,0,1\}^{{\mathbb{N}^*}}$, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.

keywords: Chowla conjecture Sarnak conjecture Möbius orthogonality ergodic theory theory of joinings
DCDS
Spectral multiplicities for ergodic flows
Alexandre I. Danilenko Mariusz Lemańczyk
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 4271-4289 doi: 10.3934/dcds.2013.33.4271
Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.
keywords: Ergodic flow spectral multiplicities.
DCDS
A class of mixing special flows over two--dimensional rotations
Krzysztof Frączek Mariusz Lemańczyk
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 4823-4829 doi: 10.3934/dcds.2015.35.4823
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\mathbb{T}^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition \[\int_{\mathbb{T}^2}f_x(x,y)\,dx\,dy\neq 0\quad\text{ and }\quad \int_{\mathbb{T}^2}f_y(x,y)\,dx \,dy\neq 0.\] For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the mixing property is proved to hold.
keywords: special flows Measure-preserving flows stretching. von Neumann's condition mixing

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