Alexandre I. Danilenko
It is shown that if $E$ is any subset of $\Bbb N$ such that either $1\in E$ or $2\in E$ then there is a mixing transformation whose set of spectral multiplicities is $E$.
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We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing power weakly mixing infinite measure preserving transformations, (iii) construct mixing Poissonian automorphisms with a simple spectrum, etc.
For any set $M$ of natural numbers there are mixing Gaussian automorphisms and non-mixing Gaussian automorphisms with singular spectrum (as well as some automorphisms which are disjoint from all Gaussian actions) such that $ M\cup\{\infty\}$ is the set of their spectral multiplicities. We show also that for a Gaussian flow $\{G_t\}$ the sets of spectral multiplicities for some automorphisms $G_t$, $t>0$, could be different.
Given a transformation $T$ of a standard measure space $(X,μ)$, let $\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,μ)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in this survey paper which subsets of $\Bbb N\cup\{\infty\}$ are realizable as $\Cal M(T)$ for various $T$: ergodic, weakly mixing, mixing, Gaussian, Poisson, ergodic infinite measure preserving, etc. The corresponding constructions are considered in detail. Generalizations to actions of Abelian locally compact second countable groups are also discussed.
It is shown that each subset of positive integers that contains 2 is realizable as the set of essential values of the multiplicity function for the Koopman operator of some weakly mixing transformation.
For totally ergodic Z^2-actions a collection of weak limits provide the set {2,4, ..., 2 ^ n} of spectral multiplicities for their tensor product. Our conditions allow to obtain a similar result for mixing actions via some limit procedure.
Each subset $E\subset\Bbb N$ is realized as the set of essential values of the multiplicity function for the Koopman operator of an ergodic conservative infinite measure preserving transformation.
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.
We study the behavior of maximal multiplicities $mm (R^n)$ for the powers of a weakly mixing automorphism $R$. For some special infinite set $A$ we show the existence of a weakly mixing rank-one automorphism $R$ such that $mm (R^n)=n$ and $mm(R^{n+1}) =1$ for all $n\in A$. Moreover, the cardinality $cardm(R^n)$ of the set of spectral multiplicities for $R^n$ is not bounded. We have $cardm(R^{n+1})=1$ and $cardm(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: $mm(R^{n}) =n$ for $n=1,2,3,..., 2009, 2010$ but $mm(T…
Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space, let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman unitary representation of G associated with T. In the case when G is either a discrete countable Abelian group or R^n, n>0, it is shown that the sets of the form {p,q,pq}, {p,q,r,pq,pr,qr,pqr} etc. or any multiplicative (and additive) subsemigroup of N are realizable as M(T) for a weakly mixing G-action T.
We prove mixing on a general class of rank-one transformations containing all known examples of rank-one mixing, including staircase transformations and Ornstein's constructions, and a variety of new constructions.
We prove the mixing for all staircase transformations satisfied the conditions $r_j/h_j\to 0$ + $r_j\to\infty$.
We prove that irreducible, linearly recurrent, type W interval exchange transformations are always mild mixing. For every irreducible permutation the set of linearly recurrent interval exchange transformations has full Hausdorff dimension.
For different classes of measure preserving transformations, we investigate collections of sets that exhibit the property of lightly mixing. Lightly mixing is a stronger property than topological mixing, and requires that a lim inf is positive. In particular, we give a straightforward proof that any mildly mixing transformation T has a dense algebra C such that T is lightly mixing on C. Also, we provide a hierarchy for the properties of lightly mixing, sweeping out and uniform sweeping out for dense collections, and dense algebras of sets. As a result, stronger mixing realizations are given fo…
We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.
We prove that a class of infinite measure preserving transformations, satisfying a "strong" weak mixing condition, generates all rigidity sequences of all conservative ergodic invertible measure preserving transformations defined on a Lebesgue $σ$-finite measure space.
We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $ε$-over-recurrence and show that given $ε> 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $ε$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigi…
We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.
We prove that for any compact connected Lie group G and a typical interval exchange transformation T, not isomorphic to a rotation, the skew product of T with a typical G-valued function, constant on the intervals, is weakly mixing.
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.
For any infinite zero-density integer set M, we found a rigid measure-preserving transformation mixing along M by answering Bergelson's question. Gaussian and Poisson suspensions over infinite constructions are suggested as suitable examples. We also discuss rigid rank one transformations of finite measure that mix along a given set, and some analogues for group actions.
A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems.
We study mixing properties of generalized $T, T^{-1}$ transformation. We discuss two mixing mechanisms. In the case the fiber dynamics is mixing, it is sufficient that the driving cocycle is small with small probability. In the case the fiber dynamics is only assumed to be ergodic, one needs to use the shearing properties of the cocycle. Applications include the Central Limit Theorem for sufficiently fast mixing systems and the estimates on deviations of ergodic averages.
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.
J.-P. Thouvenot and the author showed via different approaches that the centralizer of a mixing rank-one infinite measure preserving transformation was trivial. In this note the author presents his joining proof. We also consider constructions with algebraic spacers as well as a class of Sidon constructions to produce new examples of mixing rank one transformations. In connection with Gordin's question on the existence of homoclinic ergodic actions for a zero entropy system we also discuss Poisson suspensions of some modifications of Sidon rank one constructions.
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.
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.
We construct a rank one infinite measure preserving transformation $T$ such that for all sequences of nonzero integers $\{k_{1},..., k_{r}\}$, $T^{k_{1}}\times...\times T^{k_{r}}$ is ergodic.
We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed direction in an L-shaped polygon are topologically mixing.
In paper [1] S. V. Tikhonov introduced a complete metric on the space of mixing transformations. It generates a topology called leash-topology. In [2] Tikhonov states the following problem: for what mixing transformation T its conjugacy class is dense in the space of mixing transformations with leash-topology? We show that it is true for every mixing T. As a corollary we get that generic mixing is rank-1.
Smooth mixing transformations with loosely Bernoulli cartesian product
本文提出与混合相容的零熵系统松散 Bernoulli 判据，并证明存在光滑混合零熵松散 Bernoulli 变换，其自笛卡尔积仍为松散 Bernoulli。
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。
In the class of Ornstein transformations the mixing property satisfies a 0-1 law. Here we consider Ornstein's construction with bounded cutting parameter. In fact, these latter transformations are not mixing, however it is proved that the weak mixing property occurs with probability one. Our situation is similar to the case of interval exchange transformations whose link with the cutting and stacking construction relies in a dynamical process called Rauzy induction.
Facteurs disjoints des transformations melangeantes
本文证明了标准概率空间中任何非混合自同构都存在一个与所有混合自同构不相交的因子，揭示了非混合系统与混合系统之间的结构分离性质。
Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates o…
We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of any genus are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations.
A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergodic extensions of a fixed, but arbitrary, ergodic transformation $T_0$. We then use a result of Connes, Feldman and Weiss to extend this relative theorem to the general (countable) amenable group.
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。
We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok-Stepin, Veech, and Avila-Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have…
We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb{N} \to \mathbb{N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty$, that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$. This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.
This paper uses a construction of M. Keane to show that there exists a topologically mixing interval exchange transformation.
We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integrals with random integrands.
We establish that there are non-mixing maps that are mixing on appropriate sequences including sequences $(s_i)$ which satisfy the Rajchman dissociated property. Our examples are based on the staircase rank one construction, $M$-towers constructions and the Gaussian transformations. As a consequence, we obtain there are non-mixing maps which are mixing along the squares. We further prove that a sequence $M=(m_n)$ is a mixing sequence for some weak mixing ${1}/{2}$-rigid transformation $T$ if and only if the complement of $M$ is a thick set. This result is generalized to ${r}/{(r+1)}$-rigid tra…
Keplerian shear for Chacon Transformations
本文提出了一种适用于弱混饨系统的开普勒剪切方法，并将其应用于一类秩为一的查康变换，证明了该系统具有开普勒剪切性质。
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…
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 studi…
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.
This text is addressed to students. It is a short story about some problems in ergodic theory, both related and independent. We discuss the factorization of transformations into the product of three involutions; Furstenberg's theorem on weak multiple mixing in the mean along progressions; ergodic Furstenberg-Roth's theorem. In connection with Ledrappier's $Z^2$-action, we consider harmonic $Z_2$-sequences on $Z^2$ and experimentally observe the dominant infinite threads arising in them. The property of multiple mixing for systems with the Gordin ergodic group is proven. We discuss the absence …
The following generalizations of the Chacon map are proposed: instead of classical constant spacer sequence $(0,1,0)$ let a sequence $(0,s_j,0)$ be one with unbounded $s_j$. (We mention also an analogue of the historical Chacon map with spacer sequences in the form $(0,s_j)$.) This narrow class of rank-one transformations may be abundant source of open questions. All such constructions have partial rigidity, but some other properties could be different. For root sequence, $ s_j= [\sqrt{j}]$, (or $ s_j= [\ln{j}]$) the corresponding action is rigid, moreover it possesses all polynomials in its w…
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…
A rank one mild mixing system without minimal self joinings
本文证明存在一个温和混合但没有极小自连接的秩1变换，回答了Thouvenot提出的问题。
We discuss multiple versions of rational ergodicity and rational weak mixing for "nice" transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.
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…
We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of interm…
Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show that \[ \{0<t<1\mid f_{t} \text{is weakly mixing}\} \] is a residual subset of $X$ of full Lebesgue measure. The result is proved by establishing a generic Diophantine sufficient condition on $t$ for $f_{t}$ to be weakly mixing.
We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Holder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is non-uniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simpl…
We construct a $C^r$ transformation of the interval (or the torus) which is topologically mixing but has no invariant measure of maximal entropy. Whereas the assumption of $C^{\infty}$ ensures existence of maximal measures for an interval map, it shows we cannot weaken the smoothness assumption. We also compute the local entropy of the example.
We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_α$ using the full parameter range $0< α< \infty$, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest $α$) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of retur…
Tied-down occupation times of infinite ergodic transformations
该研究针对一类弱混合逐点对偶遍历变换，证明其游程后约束时刻停留时间的分布极限定理，提出约束更新混合性质以细化有理弱混合，并建立Gibbs-Markov与AFU映射的周期局部极限定理。
Size of exceptional sets in weakly mixing systems
本文研究了Chacon变换及一类称为限制紧映射的切割-堆叠变换的例外集，构造了统一的例外集J，并证明了其在对数尺度下的有效性，同时探讨了弱混合模型中的例外集障碍问题。
Limit theorems for low dimensional generalized $(T,T^{-1})$ transformations
针对底映射满足多重混合局部极限定理等条件、纤维为1或2维指数混合$\bbR^d$作用的广义$(T,T^{-1})$变换，证明零漂移斜值上循环下遍历和的极限定理与随机环境随机行走一致，补充了此前经典CLT的相关研究。
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