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Zemer Kosloff

Publications and source records attributed to Zemer Kosloff.

At least 19 recordsLinked to original sources
arXiv2026

Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry

We study the two $\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\ell^{2}$-Dirichlet subspaces of $\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\mathcal{D}_{2}\left(G,λ\right)=\mathcal{D}_{2}\left(G,ρ\right)\quad\Longleftrightarrow\quad G \text{ is an FC-group}.$$ The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.

math.GR↗
arXiv2024

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

We show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular Bernoulli shift satisfies the Doeblin condition, and has a limiting marginal distribution p, then it cannot have, as a finitary factor, an independent and identically distributed (iid) system of entropy larger than H(p); on the other hand, we show that iid systems with entropy strictly lower than H(p) can be obtained as finitary factors of these Bernoulli shifts, extending Keane and Smorodinsky's finitary version of Sinai's factor theorem to the nonsingular setting. As a consequence of our results we also obtain that every transitive twice continuously differentiable Anosov diffeomorphism on a compact manifold, endowed with volume measure, has iid factors, and if the factor is required to be finitary, then the iid factor cannot have Kolmogorov-Sinai entropy greater than the measure-theoretic entropy of a Sinai-Ruelle-Bowen measure associated with the Anosov diffeomorphism.

math.DS↗
arXiv2024

Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the $2$-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.

math.DS↗
arXiv2023

Stable Functional CLT for deterministic systems

We show that alpha stable Lévy motions can be simulated by any ergodic and aperiodic probability preserving transformation. Namely we show: - for $0<α<1$ and every $α$ stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$. - for $1\leq α<2$ and every symmetric alpha stable Lévy motion $\mathbb{W}$, there exists a function f whose partial sum process converges in distribution to $\mathbb{W}$, - for $1< α<2$ and every $-1\leqβ\leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_α(\ln(2), β,0)$ random variable.

math.DS↗
arXiv2023

Stable CLT for deterministic systems

We show that for every ergodic and aperiodic probability preserving transformation and $α\in (0,2)$ there exists a function whose associated time series is in the standard domain of attraction of a non-degenerate symmetric $α$-stable distribution.

math.DS↗
arXiv2021

Some factors of nonsingular Bernoulli shifts

We give elementary constructions of factors of nonsingular Bernoulli shifts. In particular, we show that all nonsingular Bernoulli shifts on a finite number of symbols which satisfy the Doeblin condition have a factor that is equivalent to an independent and identically distributed system. We also prove that there are type-III:1 Bernoulli shifts of every possible ergodic index, answering a question of Danilenko and Lemanczyk (Ergodic Theory Dynam. Systems, 39(12):3292-3321, 2019).

math.DS↗
arXiv2021

Krieger's type of nonsingular Poisson suspensions and IDPFT systems

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

math.DS↗
arXiv2020

Nonsingular Poisson Suspensions

The classical Poisson functor associates to every infinite measure preserving dynamical system $(X,μ,T)$ a probability preserving dynamical system $(X^*,μ^*,T_*)$ called the Poisson suspension of $T$. In this paper we generalize this construction: a subgroup Aut$_2(X,μ)$ of $μ$-nonsingular transformations $T$ of $X$ is specified as the largest subgroup for which $T_*$ is $μ^*$-nonsingular. Topological structure of this subgroup is studied. We show that a generic element in Aut$_2(X,μ)$ is ergodic and of Krieger type III$_1$. Let $G$ be a locally compact Polish group and let $A:G\to\text{Aut}_2(X,μ)$ 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 $κ$ on $G$ and $t>0$, a nonsingular Poisson $G$-action is constructed whose Furstenberg $κ$-entropy is $t$.

math.DS↗
arXiv2020

Ergodicity and type of nonsingular Bernoulli actions

We determine the Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (\{0,1\},μ_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $μ_g$. We prove in particular that the action is never of type II$_\infty$ if $G$ is abelian and not locally finite, answering Krengel's question for $G = \mathbb{Z}$. When $G$ is locally finite, we prove that type II$_\infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$_1$. When $G$ has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nontrivial first $L^2$-cohomology.

math.DS↗
arXiv2020

Finitary isomorphisms of Brownian motions

Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow and thus Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h >0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0, qh] for all positive rationals q.

math.DS↗
arXiv2020

Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1

We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale systems of the form \[ x_{k+1} = x_k + n^{-1} a_n(x_k,y_k) + n^{-1/2} b_n(x_k,y_k), \quad y_{k+1} = T_n y_k, \] where the fast dynamics is given by a family $T_n$ of nonuniformly expanding maps. Part 1 builds on our recent work on martingale approximations for families of nonuniformly expanding maps. We prove an iterated weak invariance principle and establish optimal iterated moment bounds for such maps. (The iterated moment bounds are new even for a fixed nonuniformly expanding map T.) The homogenization results are a consequence of this together with parallel developments on rough path theory in Part 2 by Chevyrev, Friz, Korepanov, Melbourne & Zhang.

math.DS↗
arXiv2020

Generic nonsingular Poisson suspension is of type $III_1$

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

math.DS↗
arXiv2019

Local limit theorem in deterministic systems

We show that for every ergodic and aperiodic probability preserving system, there exists a $\mathbb{Z}$ valued, square integrable function $f$ such that the partial sums process of the time series $\left\{f\circ T^i\right\}_{i=0}^\infty$ satisfies the lattice local limit theorem.

math.DS↗
arXiv2018

Boundary of the Range of a random walk and the Fölner property

The range process $R_n$ of a random walk is the collection of sites visited by the random walk up to time $n$. In this work we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Fölner sequence and show the following results: % (a) The size of the inner boundary $|\partial R_n|$ of the range of recurrent aperiodic random walks on $\mathbb{Z}^2$ with finite variance and aperiodic random walks in $\mathbb{Z}$ in the standard domain of attraction of the Cauchy distribution, divided by $\frac{n}{\log^2(n)}$, converges to a constant almost surely. % (b) We establish a formula for the Fölner asymptotic of transient cocycles over an ergodic probability preserving transformation and use it to show that for transient random walk on groups which are not virtually cyclic, for almost every path, the range is not a Fölner sequence. % (c) For aperiodic random walks in the domain of attraction of symmetric $α$- stable distributions with $1<α\leq 2$, we prove a sharp polynomial upper bound for the decay at infinity of $|\partial R_n|/|R_n|$. This last result shows that the range process of these random walks is almost surely a Fölner sequence.

math.PR↗
arXiv2018

Proving ergodicity via divergence of ergodic sums

A classical fact in ergodic theory is that ergodicity is equivalent to almost everywhere divergence of ergodic sums of all nonnegative integrable functions which are not identically zero. We show two methods, one in the measure preserving case and one in the nonsingular case, which enable one to prove this criteria by checking it on a dense collection of functions and then extending it to all nonnegative functions. The first method is then used in a new proof of a folklore criterion for ergodicity of Poisson suspensions which does not make any reference to Fock spaces. The second method which involves the double tail relation is used to show that a large class of nonsingular Bernoulli and inhomogeneous Markov shifts are ergodic if and only if they are conservative. In the last section we discuss an extension of the Bernoulli shift result to other countable groups including $\mathbb{Z}^{d},\ d\geq 2$ and discrete Heisenberg groups.

math.DS↗