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### Open Access Journals

JMD

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.

DCDS

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

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.

DCDS

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.

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