Ergodic Theory: Nonsingular Transformations

Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(μ_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}μ_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,μ_g)$ 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 n…

Nonsingular transformations and dimension spaces

For any adic transformation $T$ defined on the path space $X$ of an ordered Bratteli diagram, endowed with a Markov measure $μ$, we construct an explicit dimension space (which corresponds to a matrix values random walk on $\mathbb{Z}$) whose Poisson boundary can be identified as a $\mathbb{Z}$-space with the dynamical system $(X,μ,T)$. We give a couple of examples to show how dimension spaces can be used in the study of nonsingular transformations.

Nonsingular Group Actions and Stationary S alpha S Random Fields

This paper deals with measurable stationary symmetric stable random fields indexed by R^d and their relationship with the ergodic theory of nonsingular R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish extensions of some structure results of stationary S alpha S processes to SαS fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.

Effects of randomization on asymptotic periodicity of nonsingular transformations

It is known that the Perron--Frobenius operators of piecewise expanding $\mathcal{C}^2$ transformations possess an asymptotic periodicity of densities. On the other hand, external noise or measurement errors are unavoidable in practical systems; therefore, all realistic mathematical models should be regarded as random iterations of transformations. This paper aims to discuss the effects of randomization on the asymptotic periodicity of densities.

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).

Strict Doubly Ergodic Infinite Transformations

We give examples of rank-one transformations that are (weak) doubly ergodic and rigid (so all their cartesian products are conservative), but with non-ergodic $2$-fold cartesian product. We give conditions for rank-one infinite measure-preserving transformations to be (weak) doubly ergodic and for their $k$-fold cartesian product to be conservative. We also show that a (weak) doubly ergodic nonsingular group action is ergodic with isometric coefficients, and that the latter strictly implies W measurable sensitivity.

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine…

Subsequence Rational Ergodicity of Rank-One Transformations

We show that all rank-one transformations are subsequence boundedly rationally ergodic and that there exist rank-one transformations that are not weakly rationally ergodic.

Ergodic Properties of the Random Walk Adic Transformation over the Beta Transformation

We define a random walk adic transformation associated to an aperiodic random walk on $G=\mathbb{Z}^{k}\times\mathbb{R}^{D-k}$ driven by a $β$-transformation and study its ergodic properties. In particular, this transformation is conservative, ergodic, infinite measure preserving and we prove that it is asymptotically distributionally stable and bounded rationally ergodic. Related earlier work appears in [AS] and [ANSS] for random walk adic transformations associated to an aperiodic random walk driven by a subshift of finite type.

遍历变换讲义

Lecture notes on ergodic transformations

本讲义系统介绍了遍历变换的核心理论，涵盖遍历性等价性质、Birkhoff定理、弱混合、典型熵不变量、Poisson悬浮及酉算子谱定理等关键主题。

On the Hausdorff dimensions of a singular ergodic measure for some minimal interval exchange transformations

We show some results about the Hausdorff dimension of particular minimal but not uniquely ergodic interval exchange transformations. There is an appendix which shows that typical points for two different ergodic measures of an interval exchange can approximate each other differently.

Convergence of multiple ergodic averages along cubes for several commuting transformations

We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently by B. ~Host.

Rank-one flows of transformations with infinite ergodic index

A rank-one infinite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that for each $t\ne 0$, the Cartesian powers of the transformation $T_t$ are all ergodic.

Nonsingular $α$-rigid maps: Short proof

It is shown that for every $α$, where $α\in [0, 1/2]$, there exists an $α$-rigid transformation whose spectrum has Lebesgue component. This answers the question posed by Klemes and Reinhold in [7]. We apply a certain correspondence between weak limits of powers of a transformation and its skew products.

Convergence of multiple ergodic averages for some commuting transformations

We prove the $L^{2}$ convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, ..., T_{l}$, assuming that each map $T_i$ and each pair $T_iT_j^{-1}$ is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

Ergodic Volterra Quadratic Transformations of Symplex

In the paper a Volterra quadratic stochastic operators of three dimensional simplex into itself is considered.The full description of ergodic properties such operators is given.

Ergodicity and Conservativity of products of infinite transformations and their inverses

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

On Infinite Transformations with Maximal Control of Ergodic Two-fold Product Powers

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset $R\subset \mathbb Q\cap (0,1)$ there exists $T$ in this class such that $T^p\times T^q$ is ergodic if and only if $\frac{p}{q} \in R$. This contrasts with the finite measure preserving case where $T^p\times T^q$ is ergodic for all nonzero $p$ and $q$ if and only if $T\times T$ is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer $k$, a fa…

Mixing on Rank-One Transformations

We prove that mixing on rank-one transformations is equivalent to the spacer sequence being slice-ergodic. Slice-ergodicity, introduced in this paper, generalizes the notion of ergodic sequence to the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums. We show that polynomial staircase transformations satisfy this condition and therefore are mixing.

Mixing on a class of rank one transformations

We prove that a rank one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences in the sense that the mean ergodic theorem holds for what we call dynamical sequences. In particular, Adams' class of staircase transformations and Ornstein's class constructed using ``random spacers'' have both restricted growth and uniformly ergodic spacer sequences.

Hausdorff dimension for ergodic measures of interval exchange transformations

I show that there exist minimal interval exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. I will also show that in particular cases one can bound the Hausdorff dimension between $\frac 1 {2r+4}$ and $\frac 1 r$ for any r greater than 1.

存在一个非唯一遍历的区间交换变换具有翻转并具有三个不变测度

Existence of a Non-Uniquely Ergodic Interval Exchange Transformation with Flips Possessing Three Invariant Measures

本文首次给出了一个具有翻转的区间交换变换（FIET）具有三个不同的不变遍历测度的例子，通过推广M. Keane的方法，利用适应于FIET的Rauzy归纳法，推动了该类动力系统遍历性质的研究。

Random permutations and unique fully supported ergodicity for the Euler adic transformation

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey

This is a self-contained introduction to the applications of ergodic theory of nonsingular (also known as quasi-invariant) group actions and the structure theorem for finitely generated abelian groups on the extreme values of stationary symmetric stable random fields indexed by $\mathbb{Z}^d$. It is based on a mini course given in the Eighth Lectures on Probability and Stochastic Processes (held in the Bangalore Centre of Indian Statistical Institute during December 6-10, 2013) except that a few recent references have been added in the concluding part. This article is a survey of existing work…

On conservative sequences and their application to ergodic multiplier problems

The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb{Z}$ such that $T^n A \cap A \not = \varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\{T_i\}$, there exists a rank-one transformation $S$ such that $T_i \times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ …

On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations

We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.

Ergodic seminorms for commuting transformations and applications

Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide some tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they will be useful in the solution of other problems.

On ergodic transformations that are both weakly mixing and uniformly rigid

We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties. We show that on Cantor space, there does not exist a finite measure-preserving, totally ergodic, uniformly rigid transformation. We briefly discuss general group actions and show that (measurable) weak mixing and uniform rigidity can coexist in a more general setting.

Infinite symmetric ergodic index and related examples in infinite measure

We define an infinite measure-preserving transformation to have infinite symmetric ergodic index if all finite Cartesian products of the transformation and its inverse are ergodic, and show that infinite symmetric ergodic index does not imply that all products of powers are conservative, so does not imply power weak mixing. We provide a sufficient condition for $k$-fold and infinite symmetric ergodic index and use it to answer a question on the relationship between product conservativity and product ergodicity. We also show that a class of rank-one transformations that have infinite symmetric …

On some generic classes of ergodic measure preserving transformations

We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.

关于非唯一测度族的Hausdorff维数

On the Hausdorff Dimension of Measures for a Non-Uniquely Ergodic Family of Interval Exchange Transformations

本文构造了一个具有最大可能测度数的非唯一测度区间交换变换族，并推广了对M.Keane例子中两个测度的Hausdorff维数估计结果。

不可数对一变换的后向遍历定理

A Backward Ergodic Theorem for Uncountable-to-one Transformations

通过去除可数对一假设并利用Jankov-von Neumann一致化定理及Borel与测度场结构，将Anush Tserunyan和Jenna Zomback的2024年后向遍历定理推广到任意保测变换。

与给定遍历变换 Shannon 轨道等价的弱混合变换

Weak Mixing Transformation Which Is Shannon Orbit Equivalent to a Given Ergodic Transformation

本文证明每个遍历变换都 Shannon 轨道等价于一个弱混合变换，通过改编 Fieldsteel 和 Friedman 的 Rokhlin 塔列置换构造，并进一步给出获得 Shannon 轨道等价的 Z² 作用的灵活方法。

Ergodic Properties of a Class of Discrete Abelian Group Extensions of Rank-One Transformations

We define a class of discrete abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.

对具有相互依赖多项式迭代的交换变换的ergodic平均的seminorm控制

Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates

本文研究了具有相互依赖多项式迭代的交换变换的ergodic平均，证明其受Gowers-Host-Kra半范数控制，并确定了联合ergodic的条件。

Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I.Assani in the former case, and extending results of B.Host-B.Kra and A.Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory,…

On uniform convergence in ergodic theorems for a class of skew product transformations

Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space. The main aim of this paper is to establish u…

对称区间交换变换的无限扩张的遍历性质

Ergodic properties of infinite extension of symmetric interval exchange transformations

本文研究了对称区间交换变换的无限扩张的遍历性质，证明了特定余循环的斜积在遍历基下是遍历的，并在唯一遍历性假设下推广到更一般的函数，最后通过弱混合情形得到无限遍历指数。

Actions of finite rank: weak rational ergodicity and partial rigidity

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 ergod…

Entropy and the Uniform Mean Ergodic Theorem for a Family of Sets

We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if $\mathcal{C}$ has zero entropy. When the entropy of $\mathcal{C}$ is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly str…

在花状集、区间交换变换和遍历优化上的扩张映射

Expanding Maps on Flowers, Interval Exchange Transformations, and Ergodic Optimization

研究花状集的扩张映射性质，探讨其与区间交换变换的关系，并通过数值结果支持三角多项式在花状集上最大化的猜想。

Diophantine approximation with nonsingular integral transformations

Let $Γ$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\leq p \leq n-1$ and $V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise action of $Γ$ on $V$. Let $\bx\in V$ be such that $Γ\bx$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $\by \in V$ by the orbit $\{ γ\bx \mid γ\in Γ\}$, in terms of $\norm γ\norm$, and prove in particular that for all $\bx$ in the complement of a specific null set described in terms of a certain Diophantine condition, the exponent of ap…

Dynamics of infinite-multivalued transformations

We consider a transformation of a normalized measure space such that the image of any point is a finite set. We call such transformation $m$-transformation. In this case the orbit of any point looks like a tree. In the study of $m$-transformations we are interested in the properties of the trees. An $m$-transformation generates a stochastic kernel and a new measure. Using these objects, we introduce analogies of some main concept of ergodic theory: ergodicity, Koopman and Frobenius-Perron operators etc. We prove ergodic theorems and consider examples. We also indicate possible applications to …

Convergence of Polynomial Ergodic Averages of Several Variables for some Commuting Transformations

Let $(X,\mathcal{B},μ)$ be a probability space and let $T_1,..., T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1} ... T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\neq (0,...,0)$, then the averages $\frac{1}{|Φ_N|}\sum_{u\inΦ_N}\prod_{i=1}^r T_1^{p_{i1}(u)}... T_l^{p_{il}(u)}f_i$ converge in $L^2(μ)$ for all polynomials $p_{ij}\colon \mathbb{Z}^d\to\mathbb{Z}$, all $f_i\in L^{\infty}(μ)$, and all Følner sequences $\{Φ_N\}_{N=1}^{\infty}$ in $\mathbb{Z}^d$.

Infinite measure~preserving~transformations with Radon MSJ

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.

Ergodic Homoclinic Groups, Infinite Sidon Constructions and Poisson Suspensions

1. We answer Michael Gordin's question providing singular spectrum for transformations with homoclinic Bernoulli flows via Poisson suspensions induced by modified Sidon rank-one constructions. 2. We give homoclinic proof of Emmanuel Roy's theorem on multiple mixing of Poisson suspensions, adding new examples to Jonathan King's ergodic homoclinic groups of special zero-entropy transformations. 3. Sasha Prikhod'ko found the fast decay of correlations for some iceberg automorphisms. We get similar correlations for a class of infinite rank-one Sidon transformations. This version is based on "On Mi…

A shrinking target theorem for ergodic transformations of the unit interval

We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\overset{\infty}{\cap}} \, \underset{i=n}{\overset{\infty}{\cup}}\, f^{-i}(B (R_α^{i} x,b_i))$$ has full Lebesgue measure for almost every $x \in [0,1)$ and almost every $α\in [0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at $x \in [0,1)$ and $R_α: [0,1) \rightarrow [0,1)$ is rotation by $α\in [0,1)$. As a corollary, we provide partial answer to a question asked …

Occupation times of sets of infinite measure for ergodic transformations

Assume that $T$ is a conservative ergodic measure preserving transformation of the infinite measure space $(X,\mathcal{A},μ)$.We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling-Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with c.f.-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These abstract results…

Several remarks on Pascal automorphism and infinite ergodic theory

We interpret the Pascal-adic transformation as a generalized induced automorphism (over odometer) and formulate the $σ$-finite analog of odometer which is also known as "Hajian-Kakutani transformation" (former "Ohio state example"). We shortly suggest a sketch of the theory of random walks on the groups on the base of $σ$-finite ergodic theory.

Speedups of ergodic group extensions

We prove that for all ergodic extensions S_1 of a transformation by a locally compact second countable group G, and for all G-extensions S_2 of an aperiodic transformation, there is a relative speedup of S_1 that is relatively isomorphic to S_2. We apply this result to give necessary and sufficient conditions for two ergodic n-point or countable extensions to be related in this way.

关于与每个遍历系统都互不相交的系统类

On the class of systems which are disjoint from every ergodic system

本文给出Gorska等人定理的直接证明，刻画与所有遍历保测变换互不相交的保测变换类，并将结论推广到任意可数作用群。

Spatial limit theorem for interval exchange transformations

We prove a spatial limit theorem for generic interval exchange transformations (IETs): for a generic IET the normalized ergodic sums of a sufficiently regular (e.g., Lipschitz) function have the same asymptotic behavior of distributions as the behaviour of ergodic integrals for a generic translation flow on a flat surface, described by A. Bufetov.

Distributional limits of positive, ergodic stationary processes and infinite ergodic transformations

In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable- and $α$-rationally ergodic tranformations are presented.

Ergodic behaviour of nonconventional ergodic averages for commuting transformations

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to the space average, of diagonal measures is obtained and we give different result according to the classification of transformations. Additionally, on the torus with special rotation. we can not only get the convergence in T.Tao's paper for every point in Td, but also get a beautiful result for ergodic behaviour.