New limit properties for infinite measure preserving systems

动力系统是用微分方程或迭代映射来描述时间演化的数学模型。随机过程是对一系列偶然事件的数学描述。除了简单的数学模型外，人们对动力系统保持无限测度的（强）混合知之甚少。粗略地说，“混合”测量初始信息丢失的速度，从某种意义上说，表明动力系统与独立随机过程的差异有多大。提议项目的第一部分侧重于证明随机性质，特别是连续时间动力系统的强混合，这些系统可以表示为半流的群扩展和半流的扰动版本。这些系统的灵感来自洛伦兹气体的物理模型，即粒子以散射体的模式弹跳。该领域最近的活动激增，在过去十年中取得了一些重要突破，但对于与非周期性散射体模式相对应的扰动系统，其行为仍然是一个悬而未决的问题。这些系统已经用各种方法进行了研究，但作业者更新理论的发展提供了新的进展。由于相关群是非紧致的，并且需要无限的度量，我们在开发/应用此类流的算子更新理论时看到的一个优势是，它有可能解决相应分布具有比以前更重的尾部的情况。更重要的是，拟议的研究通过研究它们的扰动版本，解决了超越群体扩展的可能性。以往关于无限测度保持系统的强混合的结果，是在满足一定的返回时间分布的正则尾变化条件下得到的。我的项目的第二个方向围绕着以下问题：在没有规律变化的情况下，强烈的混合有意义吗？在这个主题中，我们的目标是制定和证明一个版本的（强）沿子序列混合和这样一个适当版本的沿子序列维纳引理。本文第二部分的一个（长期）目标是为具有无限均值（不一定是规则变化）的更新序列提供Erdos Feller Pollard定理的解析证明（现有的证明使用概率方法）。

A dynamical system is a mathematical model describing the time evolution by differential equations or iterated mappings. A stochastic process is a mathematical description of a sequence of chance events. Apart from simple mathematical models, very little is known about (strong) mixing for dynamical systems preserving an infinite measure. Roughly speaking, 'mixing' measures how rapidly initial information is lost and in a sense, indicates how much the dynamical system differs from an independent stochastic process. The first part of the proposed project focuses on proving stochastic properties and specifically, strong mixing of continuous time dynamical systems that can be represented as group extensions of semiflows and perturbed versions of these. These systems are inspired by physical models of Lorentz gas, i.e., particles bouncing in a pattern of scatterers. This area has seen a surge of recent activity, with some important breakthroughs in the last decade, but for perturbed systems, which correspond to non-periodic patterns of scatterers, the behaviour is still a wide open question. These systems have been studied with a variety of methods, but the development of operator renewal theory provides new inroads. As the relevant groups are non-compact and require infinite measures, one strength we see in developing/applying operator renewal theory for such flows is the potential to address cases where the corresponding distributions have heavier tails than could be treated before. More importantly, the proposed research addresses the possibility to go beyond group extension, by studying their perturbed versions.Previous results on strong mixing for infinite measure preserving systems are obtained when the condition of regularly varying tails of certain return time distributions is satisfied. A second direction of my project revolves around the following question: does strong mixing make sense in the absence of regular variation? Within this topic, we aim to formulate and prove a version of (strong) mixing along subsequences and such an appropriate version of Wiener's lemma along subsequences. One of the (longer term) aims here of the second part of the proposal is to provide an analytic proof (the existing proof uses probabilistic methods) of the Erdos Feller Pollard Theorem for renewal sequences with infinite mean (not necessarily, regularly varying).