A Non Conventional Ergodic Theorem for a Nil-System

Topological Wiener-Wintner ergodic theorem with polynomial weights

For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds.

CF-Nil系统与二维遍历平均的收敛性

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。

Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups

Let Gamma be a finitely generated nilpotent group. We consider three closely related problems: (i) the asymptotic cone for an equivariant ergodic family of inner metrics on Gamma, generalizing Pansu's theorem; (ii) the limit shapes for First Passage Percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of Gamma; (iii) a sub-additive ergodic theorem over a general ergodic Gamma-action. The limiting objects are given in terms of a Carnot-Caratheodory metric on the graded nilpotent group associated to the Mal'cev completion of Gamma.

Equidistribution of singular measures on nilmanifolds and skew products

We prove that for a minimal rotation T on a 2-step nilmanifold and any measure mu, the push-forward T^n(mu) of mu under T^n tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density 1. These results strengthen Parry's result that such systems are uniquely ergodic. Extending the work of Furstenberg, we prove an analogous theorem for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper u…

自结合与pro-nilsystem因子的局部刚性

Local rigidity of self-joinings and factors of pro-nilsystems

本文通过证明nil系统的局部刚性定理，独立于Host-Kra遍历结构定理给出了pro-nilsystem因子封闭性的新证明。

Nilsystems and ergodic averages along primes

A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^p$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao we observe everywhere convergence of such averages for nilsystems and continuous functions.

通过幂零系统的局部刚性研究幂零系统的因子闭包

Factor-Closure of Pro-Nilsystems via Local Rigidity of Nilsystems

本文通过幂零系统的局部刚性结果，独立于Host-Kra结构定理证明了幂零系统的因子闭包，还推导了拓扑幂零系统的因子闭包，结合Tao的推导给出了Host-Kra结构定理的新证明。

Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems

The regionally proximal relation of order $d$ along arithmetic progressions, namely ${\bf AP}^{[d]}$ for $d\in \N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${\bf AP}^{[d]}=Δ$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. It foll…

Polynomial averages and pointwise ergodic theorems on nilpotent groups

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…

Limit theorems for higher rank actions on Heisenberg nilmanifolds

The main result of this paper is a construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, we construct \emph{Bufetov functionals} on rectangles on $(2g+1)$-dimensional Heisenberg manifolds. We prove that deviation of the ergodic integral of higher rank actions is described by the asymptotic of Bufetov functionals for a sufficiently smooth function. As a corollary, the distribution of normalized ergodic integrals which have variance 1, converges along certain subs…

On the Wiener-Wintner theorem for nilsequences

We give a Fourier analytic proof of the generalisation due to Host and Kra of the classical Wiener-Wintner theorem and give some explicit bounds on the limit of the weighted ergodic averages.

Maximal pronilfactors and a topological Wiener-Wintner theorem

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense. We show that the CF-Nil($k$) systems are precisely the class of minimal systems for which the $k$-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof we establish that pronilsystems are $coalescent$ both in the measurable and topological categories. In addition, we character…

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.

Effective equidistribution for generalized higher step nilflows

The main results of this paper are to prove bounds for ergodic averages for nilflows on general higher step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at the polynomial speed. We analyze the exponent with the speed of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. The main result follows from the technique over controlling scaling operators in irreducible representations and measure estimation on close return orbit on general nilmanifolds.

多项式增长的平滑函数下幂流形中的逐点收敛

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。

Quantitative Multiple pointwise convergence and effective multiple correlations

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.

Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages

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…

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,…

Uniformity seminorms on $\ell^\infty$ and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Z/N\Z$ introduced by Gowers in his proof of Szemerédi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemerédi's Theorem) defined by the authors. For each integer $k\geq 1$, we define seminorms on $\ell^\infty(\Z)$ analogous to these norms and seminorms. We study t…

Weighted multiple ergodic averages and correlation sequences

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the averages of this sequence times any nilsequence converge. Key role in the proof play two decomposition results of independent interest. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has …

关于 nilsystems 的最大谱型

On the maximal spectral type of nilsystems

本文通过改造 Parry 的论证，证明 $k\geq 2$ 的遍历 $k$ 步 nilsystem 的 $L^2$ 空间由离散谱部分与无限重数 Lebesgue 谱部分组成，并推广了已有 2 步及连通单连通情形的结果。

沿IP集的遍历平均与大交集性质

Ergodic averages and the large intersection property along IP sets

本文研究沿IP集的多个遍历平均，给出均值收敛与发散准则及收敛到不变投影的充分条件，证明有理障碍情形由幂零系统控制，并获逐点收敛及相关性的大交集性质。

A proof of Walsh's convergence theorem using couplings

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.

Rates of Divergence of non-Conventional Ergodic Averages

We study the rate of growth of ergodic sums along a sequence (a_n) of times: S_N f(x)=f(T^{a_1}x) + ... + f(T^{a_N}x). We characterize the maximal rate of growth of these ergodic sums and identify a number of sequences such as (2^n) that achieve this rate of growth. We also return to Khintchine's strong uniform distribution Conjecture which stated that the averages (1/N)(f(x)+f(2x mod 1)+...+f(Nx mod 1)) converge pointwise almost everywhere to \int f for an integrable function on [0,1). We give an elementary counterexample to this conjecture, showing that divergence occurs at the maximal rate.…

逐点联合遍历系统的Wiener-Wintner遍历定理的一个推广及其应用

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。

通过可测选择的非遍历Host-Kra-Ziegler结构定理在$\mathbb{Z}^d$作用上的推广

The non-ergodic Host-Kra-Ziegler structure theorem for $\mathbb{Z}^d$-actions via measurable selections

通过可测选择程序将非遍历情形归约到遍历定理，建立了保测$\mathbb{Z}^d$作用的非遍历Host-Kra-Ziegler结构定理，并给出了垂直零字符版本，该定理是第二作者与Fraczyk分类$\mathrm{ASL}_d(\mathbb{Z})$不变点过程的关键输入。

Linear sequences and weighted ergodic theorems

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form (g(S^ny)) for a measure preserving system (Y,S) and g\in L^\infty(Y), avoiding in the latter case the problem of finding the full measure set of appropriate points y.

零空间理论的范畴学方法

A Categorical Approach to Nilspace Theory

本文用范畴学方法证明零空间理论中的弱结构定理，该定理将纤维紧致立方空间表示为典范截断塔，每步为紧致阿贝尔群主丛。

Local rigidity of higher rank non-abelian action on tours

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms and prove the existence of such action on any torus TN for any even N ? 6. We also give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.

Multiple ergodic averages for flows and an application

We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages appearing in Furstenberg's proof of Szemerédi's Theorem. For each average we show that it is sufficient to prove convergence on special factors, the Host-Kra factors, which have the structure of a nilmanifold. We also give a description of the limit. In particular, if the polynomials are independent over the real numbers then the limit is the product of the integrals. We further show that if the c…

Non-conventional ergodic averages for several commuting actions of an amenable group

Let $(X,μ)$ be a probability space, $G$ a countable amenable group and $(F_n)_n$ a left Følner sequence in $G$. This paper analyzes the non-conventional ergodic averages \[\frac{1}{|F_n|}\sum_{g \in F_n}\prod_{i=1}^d (f_i\circ T_1^g\cdots T_i^g)\] associated to a commuting tuple of $μ$-preserving actions $T_1$, ..., $T_d:G\curvearrowright X$ and $f_1$, ..., $f_d \in L^\infty(μ)$. We prove that these averages always converge in $\|\cdot\|_2$, and that they witness a multiple recurrence phenomenon when $f_1 = \ldots = f_d = 1_A$ for a non-negligible set $A\subseteq X$. This proves a conjecture o…

沿$(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$的非传统双线性平均的逐点遍历定理

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。

Ergodic Theorems for coset spaces

We study in this paper the validity of the mean ergodic theorem along \emph{left} Følner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left Følner sequence in $G$, we provide examples where the \emph{mean} ergodic theorem fails in quite dramatic ways. On the other hand, if $G$ does not admit any ICC quotients, e.g. if $G$ is virtually nilpotent, then we prove that the mean ergodic theorem does indeed hold along \emph{any} left Følner sequence. In the case when a unitary representation of a countable amenable group is induced…

Pointwise ergodic theorems for non-conventional bilinear polynomial averages

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to \infty$, where $T \colon X \to X$ is a measure-preserving transformation of a $σ$-finite measure space $(X,μ)$, $P(\mathrm{n}) \in \mathbb Z[\mathrm{n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X), \ g \in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these avera…

Multiple ergodic averages in abelian groups and Khintchine type recurrence

Let $G$ be a countable abelian group. We study ergodic averages associated with configurations of the form $\{ag,bg,(a+b)g\}$ for some $a,b\in\mathbb{Z}$. Under some assumptions on $G$, we prove that the universal characteristic factor for these averages is a factor of a $2$-step nilpotent homogeneous space. As an application we derive a Khintchine type recurrence result. In particular, we prove that for every countable abelian group $G$, if $a,b\in\mathbb{Z}$ are such that $aG,bG,(b-a)G$ and $(a+b)G$ are of finite index in $G$, then for every $E\subset G$ and $\varepsilon>0$ the set $$\{g\in …

测度下沿立方体的饱和定理及其应用

Saturated theorem along cubes for a measure and applications

本文证明了极小系统沿立方体相对于极大无穷步原幂零因子的饱和点集具有全测度，并利用此结论揭示了系统有限IP独立性与遍历测度数量的关系，回答了相关猜想。

多重相关序列的非消失性

Non-vanishing of multiple correlation sequences

本文否定解决了弗兰齐基纳基斯和库卡关于幂零系统中多重相关序列消失的猜想，通过构造反例反驳了另外两个相关猜想，为此开发了\(G/\Gamma\)的傅里叶分析框架并可扩展到一般幂零系统。

定量多项式Wiener-Wintner定理

Quantitative Polynomial Wiener-Wintner Theorems

在包括幂零李群保测作用的广泛框架下，证明了定量多项式Wiener-Wintner定理，适用于遍历平均和奇异积分权重平均，证明依赖于广义多项式Carleson定理。

沿二次括号多项式的遍历平均的逐点收敛

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。

结构扩张与多重关联序列

Structured extensions and multi-correlation sequences

本文通过结合Tao(2015)方法与Furstenberg对应原理将遍历多维逆定理归约为有限性版本，证明多重关联序列可分解为广义幂零序列与零序列之和，证实Frantzikinakis猜想，还得到带拟多项式界的多维有限性逆定理。

Convergence of ergodic averages for many group rotations

Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages M_N^αf(x). The f-rotation set is Gamma_f={α\in G: M_N^α f(x) converges for m a.e. x as N\to \infty .} We prove that if G is a compact locally connected Abelian group and f: G -> R is a measurable function then from m(Gamma_f)>0 it follows that f \in L^1(G). A similar result is established for ordinary Birkhoff averages if G=Z_{p}, the group of p-adic integers. …