Amos Nevo
The survey presents the main developments obtained over the last decade regarding pointwise ergodic theorems for measure preserving actions of locally compact groups. The survey includes an exposition of the solutions to a number of long standing open problems in ergodic theory, some of which are very recent and have not yet appeared elsewhere.
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We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.
Pointwise Ergodic Theorems for Higher Levels of Mixing
本文在弱混合和强混合系统中强化了Birkhoff ergodic定理，证明了弱混合系统的点wise定理比Wiener-Wintner定理更强，并分别刻画了弱混合和强混合系统。
Amos Nevo established the pointwise ergodic theorem in $L^p$ for measure-preserving actions of $\mathrm{PSL}_2(\mathbb{R})$ on probability spaces with respect to ball averages and every $p>1$. This paper shows by explicit example that Nevo's Theorem cannot be extended to $p=1$.
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…
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。
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.
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,…
An extension of the Wiener-Wintner ergodic theorem for pointwise jointly ergodic systems and its applications
该研究将经典Wiener-Wintner遍历定理推广到带nilsequence权的逐点联合遍历系统，并给出其在加权遍历平均收敛、沿⌊αn⌋子序列遍历平均几乎处处收敛方面的应用。
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…
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.
A pointwise ergodic theorem along return times of rapidly mixing systems
该研究引入快速相关衰减系统中由返回/到达时间随机生成的稀疏序列，证明其遍历且对遍历平均逐点普遍$L^2$-良好，为Bourgain返回时间定理的自然变体，还通过例子验证并适配非独立情形技术。
We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type $III_1$. We show that this class contains all irreducible lattices in connected semisimple Lie groups without compact factors. We also establish similar results when the stable type is $III_λ$, $0 < λ< 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of p.m.p. actions of amenable gro…
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。
We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ φ_1(n)\} < ψ(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where $p > 1$ and $r > 2$.
We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesaro averages of spherical averages in a Fuchsian group.
We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired averages. This tiling problem is easy to solve for $\mathbb{Z}$ with intervals as tiles. However, it would be interesting to find other classes of groups and sequences of tiles for which this can be done, since then our approach would yield a pointwise ergodic theorem for such classes.
Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, μ, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},μ)$, any $f \in L^1(X)$, and any $n \geq 1$, define the averaging operators $${\mathcal A}_n f(x) := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x ),$$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f \in L^1(X)$ such that the sequence ${\mathcal A}_n f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maxi…
We provide an exposition of the proofs of Bourgain's polynomial ergodic theorems. The focus is on the motivation and intuition behind his arguments.
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…
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。
We prove mean and pointwise ergodic theorems for the action of a discrete lattice subgroup in a connected algebraic Lie group, on infinite volume homogeneous algebraic varieties. Under suitable necessary conditions, our results are quantitative, namely we establish rates of convergence in the mean and pointwise ergodic theorems, which can be estimated explicitly. Our results give a precise and in most cases optimal quantitative form to the duality principle governing dynamics on homogeneous spaces. We illustrate their scope in a variety of equidistribution problems.
In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the CAT(0) setting the role of ergodic averages is played by the barycentres of the empirical distributions of a CAT(0)-valued map along an orbit of the group action. The proof rests on an approximation argument and an appeal to that result for real-valued maps.
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.
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。
Uniform vector-valued pointwise ergodic theorems for operators
该研究证明了一类含遍历Koopman算子与压缩乘法算子复合的算子的一致向量值Wiener-Wintner定理，结果在复值函数情形下全新，还适用于部分非正非压缩算子，并给出几类Koopman算子的新一致逐点定理。
The circle method and pointwise ergodic theorems
本文通过圆方法探讨了多项式遍历平均算子在几乎处处收敛与范数收敛之间的本质差异，并指出不同工具可能带来更深入的洞察。
Ergodic averages along sequences of slow growth
研究沿序列 $Ω(n)$ 的加权遍历平均逐点收敛性，通过验证双重对数逐点遍历定理证明 $Ω(n)$ 非固有强扫除，并证明慢增长序列的强扫除性质在特定扰动下保持不变。
Weak-type (1,1) inequality for discrete maximal functions and pointwise ergodic theorems along thin arithmetic sets
本文建立了沿稀疏算术集合的离散极大函数弱型 (1,1) 界，由此获得 $L^1$ 逐点收敛结果作为反例，并推导出 $L^p$ 上的多参数逐点遍历定理。
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.
Under suitable hypotheses we establish a quantitative pointwise ergodic theorem which applies to trimmed Birkhoff sums of weakly integrable functions.
Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-α}$, $0 < α< 1/2$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ orthogonal to the invariant factor, the modulated, random averages \[ \sup_{b} \Big| \frac{1}{N} \sum_{n = 1}^N b(n) T^{a_{n}} f \Big| \] converge to $0$ pointwise almost everywhere, where the supremum is taken over a set of bounded functions with certain uniform approximation properties.
Pointwise ergodic theorem along primes of the form $x^2 + ny^2$
本文利用 Hardy-Littlewood 圆法证明沿形如 $x^2 + ny^2$ 的素数多项式值集合的遍历平均点态收敛，并说明该结果不能推广到 $L^1$ 函数类。
Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0 < a < 1/2$, and let $p(n) = n^{1+ε}$, $0 < ε< 1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ the modulated, random averages \[ \frac{1}{N} \sum_{n = 1}^N e(p(n)) T^{a_n(ω)} f\] converge to $0$ pointwise almost everywhere.
Pointwise convergence of double ergodic averages along certain non-polynomial sequences
本文研究双重遍历平均沿非多项式序列的逐点收敛，证明在指数 $c\in(1,23/22)$ 且系数满足条件时极限几乎处处存在，并推广到多维情形。
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.
Extending a result by Chilin and Litvinov, we show by construction that given any $σ$-finite infinite measure space $(Ω,\mathcal{A}, μ)$ and a function $f\in L^1(Ω)+L^\infty(Ω)$ with $μ(\{|f|>\varepsilon\})=\infty$ for some $\varepsilon>0$, there exists a Dunford-Schwartz operator $T$ over $(Ω,\mathcal{A}, μ)$ such that $\frac{1}{N}\sum_{n=1}^N (T^nf)(x)$ fails to converge for almost every $x\inΩ$. In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $L^1(Ω)+L^\infty(Ω)$.
We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\ldots, T_a$ on a Borel probability space of the form $$ \sum_{n=1}^N T_a^n A_{a-1}T_{a-1}^nA_{a-1}\cdot \ldots \cdot A_0 T_0^nf$$ for $f\in L^p(X,μ)$, $p\geq 1$. We also discuss examples and present a continuous version of the result.
We prove a non conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.
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 …
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $σ$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $ρ$-variational inequalities on $L^2$ for $2<ρ<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fouri…
Pointwise ergodic averages along the Omega function in number fields
研究沿Omega函数的遍历平均的强扫除性质，证明其在数域中的失败，并回答Le等人提出的问题；同时在唯一遍历系统中建立沿数域理想上Omega函数的逐点收敛，并推导数论推论。
Continuous pointwise ergodicity for semigroup actions on locally compact spaces
该研究针对局部紧空间上的半群作用，建立了逐点连续遍历性的算子理论刻画，推广了紧空间群作用的经典结果，且适用于无Følner序列的半群，还给出了相关动力学实例。
We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.
Pointwise ergodic theorems along fractional powers of primes
该研究建立了沿素数分数次幂及更一般c-正则变化函数序列的非传统遍历平均的逐点收敛性，给出多参数振荡估计与对应定理，借助指数和估计得到Waring型结果，并在Riemann zeta函数无零点带假设下探讨L^1上的条件性结果。
We discuss an invertible version of Furstenberg's `Ergodic CP Shift Systems'. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.
Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory
本研究证明了Bourgain所研究的极大傅里叶乘子算子变分形式的最优$L^p$估计，并将其实变量多项式整数部分遍历平均逐点收敛的结果推广到更宽函数类，给出了对应收敛的$p$值范围。
We prove that if $μ_n$ are probability measures on $Z$ such that $\hat μ_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to $μ_{n_k}$ satisfy a pointwise ergodic theorem in $L^1$. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along $n^2+ \lfloor ρ(n)\rfloor$ for a slowly growing function $ρ$. Under some monotonicity assumptions, the rate of growth of $ρ'(x)$ determines the existence of…
We investigate pointwise convergence of entangled ergodic averages of Dunford-Schwartz operators $T_0,T_1,\ldots, T_m$ on a Borel probability space. These averages take 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 $f\in L^p(X,μ)$ for some $1\leq p<\infty$, and $α:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ encodes the entanglement. We prove that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for a…
For any measure preserving system $(X,\mathcal{B},μ,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order $0$ and of Fejér functions, i.e., tempered functions of order $0.$ We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of…
We consider stationary stochastic dynamical systems evolving on a compact metric space, by perturbing a deterministic dynamics with a random noise, added according to an arbitrary probabilistic distribution. We prove the maximal and pointwise ergodic theorems for the transfer operators associated to such systems. The results are extensions to noisy systems of some of the fundamental ergodic theorems for deterministic systems.
We present a general new method for constructing pointwise ergodic sequences on countable groups, which is applicable to amenable as well as to non-amenable groups and treats both cases on an equal footing. The principle underlying the method is that both cases can be viewed as instances of the general ergodic theory of amenable equivalence relations.
Pointwise convergence of ergodic averages with Möbius weight
本文证明带 Möbius 权重及更一般乘法权重的多项式多重遍历平均几乎处处逐点收敛，并解决 Frantzikinakis 的相关问题。
In this paper, we establish a noncommutative analogue of Calderón's transference principle, which allows us to deduce noncommutative ergodic maximal inequalities from the special case---operator-valued maximal inequalities. As applications, we deduce dimension-free estimates of noncommutative Wiener's maximal ergodic inequality and noncommutative Stein-Calderón's maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener's pointwise ergodic theorem, we construct a dense subset on which pointwise convergence holds following a som…
Pointwise Convergence of Random Ergodic Averages at the \(L^1\) Endpoint
本文证明沿二次增长随机序列的遍历平均在 L^1 端点几乎处处收敛，结合 Christ 调和分析与二次形式随机采样，克服均匀性障碍。
Pointwise ergodic theorem for locally countable quasi-pmp graphs
该研究证明了局部可数quasi-pmp可测图的逐点遍历定理，给出遍历图的递增有限分支Borel子图序列，推导了遍历超有限子图及自由因子的存在性，采用自包含的新方法，引入图不变量、铺砌法及可见性拟序分析超有限性条件。
Let $Γ$ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action $Γ\curvearrowright (X, μ)$ and a map $f \in L^1(X, μ)$, and to compare the global average $\int f \,\mathrm{d}μ$ of $f$ to the pointwise averages $|D|^{-1} \sum_{δ\in D} f(δ\cdot x)$, where $x \in X$ and $D$ is a nonempty finite subset of $Γ$. The basic hope is that, when $D$ runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for $μ$-almost all $x$. In this paper we prove several results that refine th…
Given any amenable group $G$ (with a left Haar measure $|\cdot|$ or $dg$), we can select out a \textit{Følner subnet} $\{F_θ,θ\inΘ\}$ from any left Følner net in $G$, which is \textit{$L^\infty$-admissible}, namely, for any Borel $G$-space $(X,\mathscr{X})$ and any $φ\in L^\infty(X,\mathscr{X})$, \begin{gather*} \lim_{θ\inΘ}\frac{1}{|F_θ|}\int_{F_θ}φ(gx)dg=φ^*(x)\ \forall x\in X\quad {\textrm{and}}\quad φ^*=(gφ)^*\ \forall g\in G. \end{gather*} Moreover, if $G$ is $σ$-compact such as a locally compact second countable Hausdorff amenable group, then $φ^*\in L^\infty(X,\mathscr{X})$, $φ^*(gx)=φ^…
We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.
We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequenc…
Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$
本文证明了在概率动力系统中，沿$(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$等$c$-正则变化稀疏轨道的双线性遍历平均的逐点收敛性，这是首个具有调制不变性的确定性稀疏轨道逐点结果。
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