This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in Mierczyński et al. [18]. Two different approaches to its existence are presented. The state space X will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$, and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains an exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.

For an arbitrary countable discrete infinite group G, non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$-actions. Given a decreasing sequence 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 non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G-action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one non-singular G-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and 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.

We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$-spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

Inspired by the phase transition results for non-singular Gaussian actions introduced in [AIM19], we prove several phase transition results for non-singular Bernoulli actions. For generalized Bernoulli actions arising from groups acting on trees, we are able to give a very precise description of their ergodic-theoretical properties in terms of the Poincaré exponent of the group.

We prove the first orbit equivalence superrigidity results for actions of type III$_\unicode{x3bb} $ when $\unicode{x3bb} \neq 1$. These actions arise as skew products of actions of dense subgroups of $\operatorname {SL}(n,\mathbb {R})$ on the sphere $S^{n-1}$ and they can have any prescribed associated flow.

We prove that many, but not all, injective factors arise as crossed products by nonsingular Bernoulli actions of the group $\mathbb {Z}$. We obtain this result by proving a completely general result on the ergodicity, type and Krieger’s associated flow for Bernoulli shifts with arbitrary base spaces. We prove that the associated flow must satisfy a structural property of infinite divisibility. Conversely, we prove that all almost periodic flows, as well as many other ergodic flows, do arise as associated flow of a weakly mixing Bernoulli action of any infinite amenable group. As a byproduct, we prove that all injective factors with almost periodic flow of weights are infinite tensor products of $2 \times 2$ matrices. Finally, we construct Poisson suspension actions with prescribed associated flow for any locally compact second countable group that does not have property (T).

It is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger 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, then G admits weakly mixing Bernoulli actions of arbitrary Krieger type.

Kingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation T of a measure space $(X, \mu )$ . We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that $\mu $ and $\mu \circ T$ have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

It is shown that for a dense $G_\delta $ -subset of the subgroup of non-singular transformations (of a standard infinite $\sigma $ -finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger’s type III1.

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 non-singular infinite direct products T of transformations $T_n$ , $n\in \mathbb N$ , of finite type is studied. It is shown that if $T_n$ is mildly mixing, $n\in \mathbb N$ , the sequence of 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 technique provides a new approach to the non-singular Gaussian transformations studied recently by Arano, Isono and Marrakchi.

We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for $\mathbb {Z}^d$ -actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for the natural action of $\mathbb {Z}^d$ on a product of d measure spaces.

Eagleson’s theorem asserts that, given a probability-preserving map, if renormalized Birkhoff sums of a function converge in distribution, then they also converge with respect to any probability measure which is absolutely continuous with respect to the invariant one. We prove a version of this result for almost sure limit theorems, extending results of Korepanov. We also prove a version of this result, in mixing systems, when one imposes a conditioning both at time 0 and at time $n$.

For a non-generic, yet dense subset of $C^{1}$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^{1}$ perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$, where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case.

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.

Based on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric α-stable discrete parameter random field. We show that the power function converges to 1 as the sample-size increases to ∞ under various classes of alternatives having longer memory in the sense of Samorodnitsky (2004). Ergodic theory of nonsingular ℤd-actions plays a very important role in the design and analysis of our large sample test.

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.