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For an arbitrary countable discrete infinite group $G$, nonsingular rank-one actions are introduced. It is shown that the class of nonsingular rank-one actions coincides with the class of nonsingular $(C,F)$-actions. Given a decreasing sequence $\Gamma_1\supsetneq\Gamma_2\supsetneq\cdots$ of cofinite subgroups in $G$ with $\bigcap_{n=1}^\infty\bigcap_{g\in G}g\Gamma_ng^{-1}=\{1_G\}$, the projective limit of the homogeneous $G$-spaces $G/\Gamma_n$ as $n\to\infty$ is a $G$-space. Endowing this $G$-space with an ergodic nonsingular nonatomic measure we obtain a dynamical system which is called a nonsingular odometer. Necessary and sufficient conditions are found for a rank-one nonsingular $G$-action to have a finite factor and a nonsingular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one nonsingular $G$-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the nonsingular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the nonsingular noncommutative case from the classical finite measure preserving one: odometer actions which are not of rank one, factors of rank-one systems which are not of rank-one, however each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

For each {\it well approximable} irrational $\theta$, we provide an explicit rank-one construction of the $e^{2\pi i\theta}$-rotation $R_\theta$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $\theta$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2\pi i\theta}$. For every irrational $\theta$, two infinite $\sigma$-finite invariant measures $\mu_\theta$ and $\mu_{\theta}'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,\mu_\theta, R_\theta)$ is {\it rigid} and of rank one and $(\Bbb T,\mu_\theta', R_\theta)$ is of {\it zero type} and of rank one. The centralizer of the latter system consists of just the powers of $R_\theta$. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction., Comment: A transliteration flaw in the name of the second named author is removed

It is shown that each non-compact locally compact second countable non-(T) group $G$ possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if $G$ is amenable. If $G$ has the Haagerup property then (and only then) these actions can be chosen of 0-type. If $G$ is amenable and unimodular then $G$ has weakly mixing Bernoulli actions of any possible Krieger's type., Comment: Slightly modified version

It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $\sigma$-finite standard measure space $(X,\mu)$ admitting an exhausting $T$-F{\o}lner sequence (i.e. a sequence $(A_n)_{n=1}^\infty$ of measured subsets of finite measure such that $A_1\subset A_2\subset\cdots$, $\bigcup_{n=1}^\infty A_n=X$ and $\lim_{n\to\infty}\sup_{g\in K}\frac{\mu(T_gA_n\triangle A_n)}{\mu(A_n)}= 0$ for each compact $K\subset G$). It is also shown that a pair of groups $H\subset G$ has property (T) if and only if there is a $\mu$-preserving $G$-action $S$ on $X$ admitting an $S$-F{\o}lner sequence and such that $S\restriction H$ is weakly mixing. These refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint., Comment: Corrected version

Given an infinite countable discrete amenable group $\Gamma$, we construct explicitly sharply weak mixing nonsingular Poisson $\Gamma$-actions of each Krieger's type: $III_\lambda$, for $\lambda\in[0,1]$, and $II_\infty$. The result is new even for $\Gamma=\Bbb Z$. As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli $\Gamma$-actions and IDPFT systems of each possible Krieger's type., Comment: Corrected, more detailed version

An inhomogeneous magnetic exchange field at a superconductor/ferromagnet interface converts spin-singlet Cooper pairs to a spin-aligned (i.e. spin-polarized) triplet state. Although the decay envelope of such triplet pairs within ferromagnetic materials is well studied, little is known about their decay in non-magnetic metals and superconductors, and in particular in the presence of spin-orbit coupling (SOC). Here we investigate devices in which triplet supercurrents are injected into the s-wave superconductor Nb. In the normal state of Nb, triplet supercurrents decay over a distance of 5 nm, which is an order of magnitude smaller than the decay of spin singlet pairs due to the SOC interacting with the spin associated with triplet pairs. In the superconducting state of Nb, triplet supercurrents are not able to couple with the singlet wavefunction and thus blocked by the absence of available equilibrium states in the singlet gap. The results offer new insight into the dynamics between s-wave singlet and s-wave triplet states., Comment: 19 pages, 5 figures

It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of nonsingular infinite direct products $T$ of transformations $T_n$, $n\in\Bbb N$, of finite type (IDPFT) is studied. It is shown that if $T_n$ is mildly mixing, $n\in\Bbb N$, the sequence of the Radon-Nikodym derivatives of $T_n$ is asymptotically translation quasi-invariant and $T$ is conservative then the Maharam extension of $T$ is sharply weak mixing. This techniques provides a new approach to the nonsingular Gaussian transformations studied recently by Arano, Isono and Marrakchi., Comment: Some new references are added

It is shown that for a dense $G_\delta$-subset of the subgroup of nonsingular transformations (of a standard infinite $\sigma$-finite measure space) whose Poisson suspensions are nonsingular, the corresponding Poisson suspensions are ergodic and of Krieger's type $III_1$.

The classical Poisson functor associates to every infinite measure preserving dynamical system $(X,\mu,T)$ a probability preserving dynamical system $(X^*,\mu^*,T_*)$ called the Poisson suspension of $T$. In this paper we generalize this construction: a subgroup Aut$_2(X,\mu)$ of $\mu$-nonsingular transformations $T$ of $X$ is specified as the largest subgroup for which $T_*$ is $\mu^*$-nonsingular. Topological structure of this subgroup is studied. We show that a generic element in Aut$_2(X,\mu)$ is ergodic and of Krieger type III$_1$. Let $G$ be a locally compact Polish group and let $A:G\to\text{Aut}_2(X,\mu)$ be a $G$-action. We investigate dynamical properties of the Poisson suspension $A_*$ of $A$ in terms of an affine representation of $G$ associated naturally with $A$. It is shown that $G$ has property (T) if and only if each nonsingular Poisson $G$-action admits an absolutely continuous invariant probability. If $G$ does not have property $(T)$ then for each generating probability $\kappa$ on $G$ and $t>0$, a nonsingular Poisson $G$-action is constructed whose Furstenberg $\kappa$-entropy is $t$., Comment: Little corrections. Subsection 8.1 is completely rewritten

Let $\Gamma$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $\Gamma$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $\Gamma$.

We propose the use of a laser beam tightly focused on a superconducting strip to create a Josephson junction by photothermal effect. The critical current of this junction can be easily controlled by the laser intensity. We show that a periodic modulation of the intensity substantially changes the dynamic properties of the junction and results in the appearance of Shapiro steps without microwave radiation. The experimental realization of optically driven Josephson junctions may open a way for the ultra-fast creation and switching of complex patterns of superconducting devices with tunable geometry and current-phase relations.

Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(\mu_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}\mu_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,\mu_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of $G$., Comment: A gap in the proof of Theorem 2.3 is corrected. An appendix is added

Let $G$ be a discrete countable infinite group. We show that each topological $(C,F)$-action $T$ of $G$ on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and R{\o}rdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying $(C,F)$-parameters. If $G$ is linearly ordered Abelian then the topological centralizer of $T$ is trivial. If $G$ is monotileable and amenable, denote by ${\cal A}_G$ the set of all probability preserving actions of $G$ on the unit interval with Lebesgue measure and endow it with the natural topology. We show that the set of $(C,F)$-parameters of all $(C,F)$-actions of $G$ furnished with a suitable topology is a model for ${\cal A}_G$ in the sense of Forman, Rudolph and Weiss. If $T$ is a rank-one transformation with bounded sequences of cuts and spacer maps then we found simple necessary and sufficient conditions on the related $(C,F)$-parameters under which (i) $T$ is rigid, (ii) $T$ is totally ergodic. It is found an alternative proof of Ryzhikov's theorem that if $T$ is totally ergodic and non-rigid rank-one map with bounded parameters then $T$ has MSJ. We also give a more general version of the criterium (by Gao and Hill) for isomorphism and disjointness of two commensurate non-rigid totally ergodic rank-one maps with bounded parameters. It is shown that the rank-one transformations with bounded parameters and no spacers over the last subtowers is a proper subclass of the rank-one transformations with bounded parameters., Comment: Extended and corrected version

It is shown that each discrete countable infinite amenable group admits a 0-entropy mixing action on a standard probability space.

Let $G$ be a discrete countable infinite group that does not have Kazhdan's property ~(T) and let $\kappa$ be a generating probability measure on $G$. Then for each $t>0$, there is a type $III_1$ ergodic free nonsingular $G$-action whose $\kappa$-entropy (or the Furstenberg entropy) is $t$.

We introduce concepts of Radon MSJ and Radon disjointness for infinite Radon measure preserving homeomorphisms of the locally compact Cantor space. We construct an uncountable family of pairwise Radon disjoint infinite Chacon like transformations. Every such transformation is Radon strictly ergodic, totally ergodic, asymmetric (not isomorphic to its inverse), has Radon MSJ and possesses Radon joinings whose ergodic components are not joinings., Comment: Final, slightly corrected, version, to appear in Isr. J. Math

Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$., Comment: Lemma 1.1(ii) is corrected. This implied little changes in the proof of Claim 1 in Theorems 0.1 and 0.3

Let $\Gamma$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $\Gamma$ and a "direction" in $G$ (i.e. an element $\theta$ of the projective space $P(\goth g)$ of the Lie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $\theta$ are introduced. It is shown that the set of recurrent directions $\Cal R(T)$ and the set of rigid directions for $T$ are both $G_\delta$. In the case where $G=\Bbb R^d$ and $\Gamma=\Bbb Z^d$, we prove that (a) for each $G_\delta$-subset $\Delta$ of $P(\goth g)$ and a countable subset $D\subset\Delta$, there is a rank-one action $T$ such that $D\subset\Cal R(T)\subset\Delta$ and (b) $\Cal R(T)=P(\goth g)$ for a generic infinite measure preserving action $T$ of $\Gamma$. This answers partly a question from a recent paper by A.~Johnson and A.~{\c S}ahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where $G$ is the Heisenberg group $H_3(\Bbb R)$ and $\Gamma=H_3(\Bbb Z)$, a rank-one $\Gamma$-action $T$ is constructed for which $\Cal R(T)$ is not invariant under the natural "adjoint" $G$-action.

Via (C,F)-construction we produce a 2-fold simple mixing transformation which has uncountably many non-trivial proper factors and all of them are prime.

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergodic Bratteli-Vershik maps corresponding to the "consequtively ordered" Bratteli diagrams of finite rank, some their generalizations and the ergodic IETs are partially rigid.