Pointwise Convergence of Multiple Ergodic Averages

遍历理论是对非常普遍的等分布现象的研究：给定一个典型的“随机化”但保持大小的有界环境X的变换T，人们试图理解轨道O(p)的性质：= {p, Tp, T^2p，…为了使这个讨论严谨，我们在X上施加测量理论结构，给它一个概率测度，即一个抽象的体积元m。从我们的数值积分工作中，我们期望如果轨道O(p)相对于m是“等分布的”，那么采样过程ref_n (p):= 1/N (f(Tp) + f(T^ 2p) +…+ f(T^N p))应该近似f对m的积分。1931年证明的George Birkhoff的逐点遍历定理的内容是，除了m可忽略的起始位置集之外，F_N(p)总是趋向于f对m的积分，前提是f的积分存在，并且T是充分随机化的。通俗地说：f的“时间平均”收敛于f的“空间平均”。遍历理论中的一个经典问题涉及Birkhoff定理的多项式推广：G_N(p)的收敛性：= 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O(P)在多项式次数限制下的均匀分布。在20世纪80年代末和90年代初，菲尔兹奖得主Jean Bourgain证明了（假设）当g有界时，除了一个m可忽略的起始位置集合外，G_N(p)也收敛；为了恢复g的积分，我们要求变换t的随机化行为稍微多一些。本文将研究多个遍历平均的收敛性，这些遍历平均是通过研究许多不同函数{h_1，…，h_m}许多不同的交换变换，{T_1，…，T_m}和许多不同阶的整数系数多项式，{P_1，…，P_m}：我们将寻求理解平均值k_n (p)的收敛性：= 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）。

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).