Christophe Cuny, Yves Derriennic
We study the almost sure convergence of bilateral ergodic averages for not necessarily integrable functions and relate it to the ones of the forward and backward averages, hence complementing results of Woś and the second named author. In the case of convergence, using results of Furstenberg on double recurrence, we prove oscillations of the bilateral ergodic averages around the limit.
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We show uniform convergence of Wiener-Wintner ergodic averages for ergodic actions of (not necessarily countable) locally compact, second countable, abelian (LCA) groups. As a by-product, we obtain a finitary version of the van der Corput inequality for such groups.
Slow convergence of ergodic averages for actions of amenable groups
研究可数 amenable 群作用的加权遍历平均，证明其收敛速度可以任意慢，揭示遍历平均的慢收敛现象。
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.
Absolute and Unconditional Convergence of Series of Ergodic Averages and Lebesgue Derivatives
该研究探讨遍历理论与调和分析中各类随机过程级数的绝对及无条件收敛条件，涉及Cesàro平均、Lebesgue导数等对象的平均差相关级数。
We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L^2 norm.
The full diapason of convergence rates of Birkhoff averages for ergodic flows
本文研究了ergodic流中Birkhoff平均收敛速率的完整范围，通过选择平均函数实现从最大速率到任意慢速率的转换，补充了Krengel关于ergodic自映射慢收敛速率的经典结果。
These notes are based on a course for a general audience given at the Centro de Modeliamento Matemático of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a product of functions taken at different times. We also describe the relations between this area of ergodic theory and some classical and some recent results in additive number theory.
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.
We study here weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in $L^{2}$. We find a necessary condition and show that for any bounded measurable function $ϕ$ on an ergodic system, the sequence $ϕ(T^{n}x)$ is universally good for almost every $x$. The linear case was understood by Host and Kra.
The well-known Jewett-Krieger's Theorem states that each ergodic system has a strictly ergodic model. Strengthening the model by requiring that it is strictly ergodic under some group actions, and building the connection of the new model with the convergence of pointwise non-conventional ergodic averages we prove that for an ergodic system $(X,\X,μ, T)$, $d\in\N$, $f_1, \ldots, f_d \in L^{\infty}(μ)$, %and any tempered F{\rm $ø$}lner sequence $F_n$ of $\Z^2$, the averages \begin{equation*} \frac{1}{N^2} \sum_{(n,m)\in F_N} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) \end{equation*} converg…
Power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in von Neumann's ergodic theorem with continuous time is considered. All possible exponents of the considered power-law convergence are found; for each of these exponents, spectral criteria for such convergence are given and a complete description of all such subspaces is obtained. Uniform convergence over the entire space takes place only in trivial cases, which explains the interest in the uniform convergence just on subspaces. In addition, along the way, the old convergence rate estimates in the vo…
Birkhoff's theorem states that for an ergodic automorphism, the time averages converge to the space average. Given sequence $ψ(n)\to+0$, U. Krengel proved that for any ergodic automorphism there is an indicator such that the corresponding time averages converged a.e. with a rate slower than $ψ$. We prove again similar statements answering a question of I. Podvigin in passing.
We apply Walsh's method for proving norm convergence of multiple ergodic averages to arbitrary amenable groups. We obtain convergence in the uniform Cesàro sense for their polynomial actions and for ``triangular'' averages associated to commuting homomorphic actions. The latter generalizes a result due to Bergelson, McCutcheon, and Zhang in the case of two actions.
In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we prove the pointwise convergence of multiple ergodic average $\disp \f 1 N \sum_…
By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},μ, T)$, $d\in{\mathbb N}$, $f_1, \ldots, f_d \in L^{\infty}(μ)$, the averages $$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) $$ converge $μ$ a.e. Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge a.e. That is, we show that if $(X,\mathcal{X},μ, T)$ is an ergodic distal system, and $f_1, \ldots, f_d \in L^{\infty}(μ)$, then multiple ergodic averages $$\frac 1 N\s…
A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},μ,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We …
It is shown that there exist a subsequence for which the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space converge almost everywhere provided that the maps are weakly mixing with an ergodic extra condition. The proof provides a example of non-singular dynamical system for which the maximal ergodic inequality does not hold. We further get that the non-singular strategy to solve the pointwise convergence of the Furstenberg ergodic averages fails.
Let $(X, \mathcal{A},μ)$ be a probability space and let $T$ be a contraction on $L^2(μ)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit $\lim\limits_{N\rightarrow\infty}\frac{1}{A_N}\sum_{k=0}^{N-1} w_k T^{u_k}(f)=0$ $μ$-a.e. for any function $f$ in $L^2(μ)$. As a consequence of our main theorems, we also deal with the so-called one-sided weighted ergodic Hilbert transforms.
Almost Uniform Convergence in Noncommutative Dunford-Schwartz Ergodic Theorem for $p>1$
本文将非交换 Dunford-Schwartz 遍历平均的收敛性从 $p=1$ 的双边几乎一致收敛推进到 $1<p<\infty$ 时的几乎一致收敛。
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,…
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…
We compute exact convergence rates in von Neumann type ergodic theorems when the acting group of measure preserving transformations is free and the means are taken over spheres or over balls defined by a word metric. Relying on the upper bounds on the spectra of Koopman operators deduced by Lubozky, Phillips, and Sarnak from Deligne's work on the Weil conjecture, we compute the exact convergence rate for the free groups (of rank $(p+1)/2$ where $p\equiv 1\mod 4$ is prime) of isometries of the round sphere defined by Lipschitz quaternions. We also show that any finite rank free group of automor…
The pointwise ergodic theorem on finitely additive spaces
本文提出适用于有限可加测度的有限几乎必然收敛概念，借此将逐点遍历定理扩展至有限可加概率空间，并通过扩展Calderón传递原理完成证明。
Weighted multiple ergodic averages via analytic observables over $ \mathbb{T}^\infty $: Is exponential pointwise convergence universal?
本文通过加权方法研究了在无限维环上多重遍历平均的指数点wise收敛性，解决了其普遍性问题，并通过反例和数值分析验证了方法的最优性。
On the speed of convergence in the ergodic theorem for shift operators
本文研究移位算子遍历平均的逐点收敛速度，证明在适当条件下其几乎处处收敛到零的速度本质上为 $N^{-1/2}$，并给出在环面自同态相关移位中的应用。
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 $\mathbb{Z}^d$.
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages \begin{equation*} \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty, \end{equation*} almost surely converge.
Slow convergence almost everywhere of ergodic averages
该研究针对$Z^n$群在概率空间上的遍历作用，证明存在可积函数使遍历平均几乎处处收敛速度可任意慢，推广了Krengel效应，借助弱化版Rokhlin引理构造渐近几乎不变集完成证明。
Pointwise Ergodic Theorems Along Reiter Sequences on Locally Compact Groups
本文针对容许Reiter序列的局部紧群，证明沿该序列的遍历平均的极大不等式，建立其双边几乎一致收敛性，并推导出非交换Wiener-Wintner型定理。
We show that effective $2\ell$-multiple correlations imply quantitative $\ell$-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.
By definition, a map quasiperiodic on a set $X$ if the map is conjugate to a pure rotation. Suppose we have a trajectory $(x_n)$ that we suspect is quasiperiodic. How do we determine if it is? In this paper we show how to compute the conjugacy map using only knowledge of $(x_n)$. Our main tool is a variant of Birkhoff averages. The Birkhoff Ergodic Theorem asserts that time averages of a function evaluated along a trajectory of length $N$ converge to the space average, the integral of $f$, as $N\to\infty$, for ergodic dynamical systems. But that convergence can be slow. Instead of uniform aver…
Polynomial ergodic averages of measure-preserving systems acted by $\mathbb{Z}^{d}$
该研究通过把一般 $\mathbb{Z}^{d}$ 作用保测系统的多项式遍历平均逐点收敛归结为零熵情形，证明了 $\mathbb{Z}^{d}$ 作用 K-系统中的相应收敛性。
For a parameterized hyperbolic system $u_{i+1} = f(u_i,s)$, the derivative of an ergodic average $\ < J\ > = \underset{n\rightarrow\infty}{\lim} \frac1n \sum_1^n J(u_i,s)$ to the parameter $s$ can be computed via the least squares sensitivity method. This method solves a constrained least squares problem and computes an approximation to the desired derivative $d\ < J\ > \over ds$ from the solution. This paper proves that as the size of the least squares problem approaches infinity, the computed approximation converges to the true derivative.
A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences
本文证明半有限冯·诺依曼代数上沿稀疏随机子序列的约当同构平均双侧几乎一致收敛，将LaVictoire随机遍历定理推广至非交换设置。
We show that for every ergodic system $(X,μ,T_1,\ldots,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \prod_{j=1}^d T_j^{n_j}x)\cdots f_d(T_d^n \prod_{j=1}^d T_j^{n_j}x). \] converges for $μ$-a.e. $x\in X$ as $N\to\infty$. If $X$ is distal, we prove that the average \[\frac{1}{N}\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\cdots f_d(T_d^nx) \] converges for $μ$-a.e. $x\in X$ as $N\to\infty$. We also establish the pointwise convergence of averages along cubical configurations…
CF-Nil systems and convergence of two-dimensional ergodic averages
本文引入CF-Nil($k$)系统并构造CF-Nil模型，证明带nilsequence权的二维多重遍历平均逐点收敛，并建立非交换变换二维平均在无零熵条件下的$L^2$收敛。
On spectral measures and convergence rates in von Neumann's Ergodic Theorem
本文研究离散系统冯·诺依曼遍历定理的收敛速率，揭示幂律衰减指数与谱测度在谱值1处逐点标度指数的关系，并证明无谱隙时时间平均速率依赖于趋于无穷的时间序列。
Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time-averages of solutions are shown.
In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.
On the rate of convergence of $\mathbb{R}^d$-ergodic averages constructed over a strictly convex set
本文针对严格凸集上构造的 $\mathbb{R}^d$ 遍历平均，利用谱判据和 Hertz 的傅里叶渐近结果，证明其收敛速率与集合具体形状无关。
We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, t\log t$ and $e^{\sqrt{\log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $\frac{1}{N} \sum_{n=1}^{N}f_1(T^{\lfloor{a_1(n)}\rfloor}x)\cdots f_k(T^{\lfloor{a_k(n)}\rfloor}x)$ exists and is equa…
We study mean ergodicity in amenable operator semigroups and establish the connection to the convergence of strong and weak ergodic nets. We then use these results in order to show the convergence of uniform families of ergodic nets that appear in topological Wiener-Wintner theorems.
We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one dimensional torus translation, these estimates are nearly optimal.
Walsh has recently proved the norm convergence of all nonconventional ergodic averages involving polynomial sequences in discrete nilpotent acting groups. He deduces this convergence from an equivalent, `finitary' assertion of stability over arbitrarily long time-intervals for these averages, which is proved by essentially finitary means. The present paper shows how the induction at the heart of Walsh's proof can also be implemented using more classical notions of ergodic theory: in particular, couplings and characteristic factors.
Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces
本文研究非交换 $L^p$-空间中由伯努利变量确定的随机稀疏子序列的遍历平均，证明其几乎处处一致收敛到遍历投影，扩展了 Bourgain 的定理。
We study double averages along orbits for measure preserving actions of $\mathbb{A}^ω$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these averages, which in particular reproves their convergence in $L^p$ for any finite $p$ and for any choice of two $L^\infty$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.
We prove a non conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.
We consider ergodic series of the form $\sum_{n=0}^\infty a_n f(T^n x)$ where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $μ$. Under certain conditions on the dynamical system $T$, the invariant measure $μ$ and the function $f$, we prove that the series converges $μ$-almost everywhere if and only if $\sum_{n=0}^\infty |a_n|^2<\infty$, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine type inequality. We also prove that the system $\{f\circ T^n\}$ is a Riesz system if and only if the spect…
We offer a generalization of the recent result of Tao (building on earlier results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages associated to an arbitrary number of commuting probability-preserving transformations always converge to some limit in L^2. We prove the corresponding result for a collection of commuting actions of a larger discrete Abelian group, and gives convergence that is uniform in the start-point of the averages. While Tao's proof rests on a conversion to a finitary problem, we i…
Let ${\bf X}=(X, Σ, m, τ)$ be a dynamical system. We prove that the bilinear series $\sideset{}{'}\sum_{n=-N}^{N}\frac{f(τ^nx)g(τ^{-n}x)}{n}$ converges almost everywhere for each $f,g\in L^{\infty}(X).$ We also give a proof along the same lines of Bourgain's analog result for averages.
Pointwise convergence of some continuous-time polynomial ergodic averages
本文基于可测流的拓扑模型研究连续时间多项式遍历平均的逐点收敛性，证明了特定条件下相关极限对几乎处处点存在，并建立了涉及测地流和水平圆周流的逐点遍历定理。
Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Ma…
We show that for each Orlicz space properly contained in L^1 there is a sequence along which the ergodic averages converge for functions in the Orlicz space, but diverge for all f in L^1. This extends the work of K. Reinhold, who, building on the work of A. Bellow,constructed a sequence for which the averages converge a.e. for every f in L^p, p>q, but diverge for some f in L^q. Our method, introduced by Bellow and extended by Reinhold and M. Wierdl, is perturbation.
We study pointwise convergence of entangled averages of the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{α(m)}}A_{m-1}T^{n_{α(m-1)}}_{m-1}\ldots A_2T_2^{n_{α(2)}}A_1T_1^{n_{α(1)}} f, \] where $f\in L^2(X,μ)$, $α:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$, and the $T_i$ are ergodic measure preserving transformations on the standard probability space $(X,μ)$. We show that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for all $f\in L^2$. We also present results for the general $L^p$ case…
We prove a.e. convergence of continuous-time quadratic averages with respect to two commuting $\mathbb{R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb{R}^2$-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.
Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen \& Cuny who showed convergence of a series along randomly perturbed times for functions in $L^2$ with $\int \max(1,\log (1+|t|)) dμ_f<\infty$. We prove universal pointwise convergence of a class of random averages along randomly perturbed times for $L^2$ functions with $\int \max(1,\log\log(1+|t|)) dμ_f<\infty$. For averages with additional smoothing properties, we obtain a universal variational inequality as well as universal pointwise conve…
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.
We investigate some ergodic and spectral properties of general (discrete) $C^*$-dynamical systems $({\mathfrak A},Φ)$ made of a unital $C^*$-algebra and a multiplicative, identity-preserving $*$-map $Φ:{\mathfrak A}\to{\mathfrak A}$, particularising the situation when $({\mathfrak A},Φ)$ enjoys the property of unique ergodicity with respect to the fixed-point subalgebra. For $C^*$-dynamical systems enjoying or not the strong ergodic property mentioned above, we provide conditions on $λ$ in the unit circle $\{z\in{\mathbb C}\mid |z|=1\}$ and the corresponding eigenspace ${\mathfrak A}_λ\subset{…
This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan, every locally constant SL(2,R)-valued cocycle is uniform. As a consequence, the corresponding Schrödinger operators exhibit Cantor spectrum of Lebesgue measure zero. An investigation of Boshernitzan's condition then shows that these results cover all earlier results of this type and, moreover, provide various new ones. In particular, Boshernitzan's condition is shown to hold for almost all circle …
We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preserving transformations, and $a_n$ is a random version of the sequence $[n^c]$ for some appropriate $c>1$. We also prove similar mean convergence results for averages of the form $\frac{1}{N}\sum_{n=1}^N f(T^{a_n}x)\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$ are powers of the same transformation. The deterministic versions of these results, where one replaces $a_n$ with $[n^c]$, remain open, and …
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