Idris Assani
Let $(X,\mathcal{B},μ, T)$ be a measure preserving system. We prove the pointwise convergence of the averages $$\frac{1}{N^2}\sum_{n,m= 0}^{N-1} f_1(T^nx)f_2(T^mx)f_3(T^{n+m}x)$$ and of similar averages with seven bounded functions.
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Let $(X,\mathcal{B},μ, T)$ be a measure preserving system. We prove the pointwise convergence of averages along cubes of $2^{k}-1$ bounded and measurable functions for all $k$.
Let $(X, \mathcal{B}, μ)$ be a probability measure space and $T_1$, $T_2$, $T_3$ three not necessarily commuting measure preserving transformations on $(X, \mathcal{B}, μ)$. We prove that for all bounded functions $f_1$, $f_2$, $f_3$ the averages $$\frac{1}{N^2}\sum_{n, m =1}^N f_1(T_1^nx)f_2(T_2^mx)f_3(T_3^{n+m}x)$$ converges a.e. Generalizations to averages of $2^k -1$ functions are also given for not necessarily commuting weakly mixing systems.
We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no …
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.
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.
Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,μ,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(μ)$.
We study the pointwise convergence of some weighted averages linked to averages along cubes. We show that if $(X,\mathcal{B},μ, T_i)$ are not necessarily commuting measure preserving systems on the same finite measure space and if $f_i,$ $1\leq i\leq 6$ are bounded functions then the averages $$\frac{1}{N^3}\sum_{n, m, p=1}^N f_1(T_1^nx) f_2(T_2^mx) f_3(T_3^px) f_4(T_4^{n+m}x) f_5(T_5^{n+p}x) f_6(T_6^{m+p}x)$$ converge almost everywhere.
We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform con…
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,…
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。
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…
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.
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 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。
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…
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 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。
Pointwise convergence in nilmanifolds along smooth functions of polynomial growth
本文研究幂流形中源自Hardy域多项式增长平滑函数的轨道等分布性，证明其在特定增长率假设下于子幂流形上一致分布，并借此结合Host-Kra结构定理与半范数估计导出多重遍历平均的范数收敛结论。
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.
It is shown that the cubic nonconventional ergodic averages of any order with a bounded aperiodic multiplicative function or von Mangoldt weights converge almost surely.
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 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 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.
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。
In 1967, Ross and Strömberg published a theorem about pointwise limits of orbital integrals for the left action of a locally compact group $G$ onto $(G,ρ)$, where $ρ$ is the right Haar measure. In this paper, we study the same kind of problem, but more generally for left actions of $G$ onto any measured space $(X,μ)$, which leaves the $σ$-finite measure $μ$ relatively invariant, in the sense that $sμ= Δ(s)μ$ for every $s\in G$, where $Δ$ is the modular function of $G$. As a consequence, we also obtain a generalization of a theorem of Civin, relative to one-parameter groups of measure preservin…
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。
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。
It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of $L^\infty$, which we generically call global obse…
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 Convergence of Sequences of Singular Measures
本研究探讨实数域上概率测度对应的卷积算子序列的几乎处处收敛性，分析近似恒等元条件下实现逐点收敛的额外要求，重点研究单测度压缩序列的相关情形。
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。
Parsings of Stationary Processes, Stopping Times and the Fundamental Pointwise Convergence Theorems of Ergodic Theory
本文引入平稳过程的单词集合解析概念及渐近分析框架，结合解析观察（基于定制停止时间的集合），从较弱的依概率收敛版本推导出逐点遍历定理与香农-麦克米兰-布莱曼定理。
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。
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^…
Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences
本文证明了在配备交换保测变换的测度空间上，沿Hardy field序列的遍历平均在 $p>1$ 时的逐点收敛性，通过建立长变分不等式和极大不等式实现，并对特定函数类给出了完全变分估计及指数和收敛速率的定量界。
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…
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…
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。
Pointwise convergence of polynomial multiple ergodic averages along the primes
研究沿素数加权的多项式多重遍历平均的点态几乎处处收敛性，通过发展多重线性圆法，结合Weyl型不等式和p进L^q改进估计，证明了收敛性并得到了尖锐的r变差估计。
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。
Pointwise convergence of bilinear polynomial averages over the primes
研究证明了在素数上的双线性多项式平均在特定条件下几乎处处收敛，并结合了多项工具和估计方法。
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。
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 $L^1(Ω)+L^\infty(Ω)$.
Pointwise behavior of SU(1,1) nonlinear Fourier transform
本文证明了SU(1,1)非线性傅里叶变换在平方可和系数下可能点态发散，并由此推出Szegö类测度的单位圆上正交多项式经典点态渐近性可能失效，同时讨论了点态收敛成立的若干特例。
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.
Pointwise convergence of ergodic averages along quadratic bracket polynomials
本文证明了沿 $(n\lfloor n\sqrt{k}\rfloor)$ 形式轨道的遍历平均的逐点收敛性，通过改进的圆法和 Green 与 Tao 的多项式轨道定量行为结果解决了超出小分母有理数的算术障碍。
Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences
该研究证明了沿Piatetski-Shapiro序列的多项式构型存在性，并确立了相关多重遍历平均的逐点几乎处处收敛性。
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