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Pointwise Convergence of Multiple Ergodic Averages

多个遍历平均值的逐点收敛

基本信息

批准号：
EP/W010275/1
负责人：
Ben Krause
金额：
$ 52.21万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW010275%2F1
关键词：
Pointwise Convergence Multiple Ergodic Averages

项目摘要

Ergodic theory is the study of equidistribution phenomena in very great generality: given a typically ``randomizing," but size preserving transformation, T, of a bounded environment, X, one seeks to understand the properties of the orbit O(p) := {p, Tp, T^2p, ... } for a typical initial position, p. To make this discussion rigorous, one imposes measure theoretic structure on X, equipping it with a probability measure, i.e. an abstract volume element, m.From our work on numerical integration, we expect that if the orbit O(p) is ``equidistributed" with respect to m, then the sampling procedureF_N(p) := 1/N (f(Tp) + f(T^2 p) + ... + f(T^N p) )should approximate the integral of f with respect to m. The content of George Birkhoff's pointwise ergodic theorem, proven in 1931, is that aside from an m-negligible set of starting locations, F_N(p) always tends towards the integral of f with respect to m, provided that the integral of f exists, and T is sufficiently randomizing. Colloquially: the "time averages" of f converge to the "space average" of f.A classical problem in ergodic theory concerns polynomial extension of Birkhoff's theorem: the convergence of G_N(p) := 1/N ( g(T^{P(1)} p) + g(T^{P(2)} p) + ... + g(T^{P(N)} p) ), where P is a polynomial with integer coefficients: i.e. the equidistribution of O(p) when restricted to polynomial times. In the late 1980s and early 1990s, the Fields Medalist Jean Bourgain proved that provided that (say) whenever g is bounded, aside from an m-negligible set of starting locations, G_N(p) also converge; in order to recover the integral of g, we require slightly more randomizing behavior from our transformation, T.This proposal will study the convergence properties of multiple ergodic averages, formed by studying interference between many different functions, {h_1,...,h_m} many different commuting transformations, {T_1,...,T_m} and many different polynomials with distinct degrees with integer coefficients, {P_1,...,P_m}: we will seek to understand the convergence of the averagesK_N(p) := 1/N( H_1(p) + H_2(p) + ... + H_N(p))whereH_n(p) := h_1(T_1^{P_1(n)} p) x ... x h_m(T_m^{P_m(n)} p).

遍历理论是对等分布现象的非常一般性的研究：给定一个有界环境X的典型的“随机化”但大小保持的变换T，人们试图理解轨道O（p）的性质：= {p，Tp，T^2p，.为使这一讨论严格，我们在X上加了测度论结构，给它配备一个概率测度，即一个抽象的体积元m.从我们的数值积分工作中，我们预期，如果轨道O（p）关于m是“等分布的”，则抽样程序F_N（p）：= 1/N（f（Tp）+ f（T^2p）+…. + f（T^N p））应该近似f关于m的积分。乔治·伯克霍夫（George Birkhoff）于1931年证明的逐点遍历定理的内容是，除了m个可忽略的起始位置集之外，F_N（p）总是倾向于f关于m的积分，前提是f的积分存在，并且T充分随机化。通俗地说：遍历理论中的一个经典问题是Birkhoff定理的多项式推广：G_N（p）：= 1/N（g（T^{P（1）} p）+ g（T^{P（2）} p）+. + g（T^{P（N）} p）），其中P是具有整数系数的多项式：即当限制为多项式时间时O（p）的等分布。在20世纪80年代末和90年代初，菲尔兹奖获得者Jean Bourgain证明了，如果（比如）当g有界时，除了起始位置的m-可忽略集之外，G_N（p）也收敛;为了恢复g的积分，我们需要来自我们的变换T的稍微更多的随机化行为。该提议将研究多个遍历平均的收敛性质，通过研究许多不同函数之间的干扰而形成，{h_1，...，h_m}许多不同的交换变换，{T_1，.，T_m}和具有整数系数的不同次数的许多不同多项式{P_1，.，P_m}：我们将试图理解平均数K_N（p）：= 1/N（H_1（p）+ H_2（p）+…+ H_N（p））其中H_n（p）：= h_1（T_1^{P_1（n）} p）x. x h_m（T_m^{P_m（n）} p）.