New limit properties for infinite measure preserving systems

动力系统是通过微分方程式或迭代映射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，关于保持无穷大测度的动力系统的(强)混合知之甚少。粗略地说，“混合”衡量了初始信息丢失的速度，在某种意义上，它表明了动力系统与独立的随机过程有多大的不同。该项目的第一部分集中于证明随机性质，特别是连续时间动力系统的强混合，它可以表示为半流的群扩张和这些扰动的形式。这些系统的灵感来自洛伦兹气体的物理模型，即粒子在散射体模式中反弹。最近这一领域的活动激增，在过去十年中取得了一些重要突破，但对于与散射体的非周期模式相对应的扰动系统来说，这种行为仍然是一个广泛的悬而未决的问题。人们已经用各种方法研究了这些系统，但算子更新理论的发展提供了新的突破。由于相关群是非紧致的，需要无限的措施，我们在开发/应用这类流的算子更新理论时看到的一个优点是，有可能解决相应分布具有比以前可以处理的更重尾部的情况。更重要的是，通过研究它们的扰动形式，提出了超越群扩张的可能性。当满足某些返回时间分布的规则变尾的条件时，得到了关于无限保测系统强混合的前人的结果。我的项目的第二个方向围绕以下问题：在没有规则变化的情况下，强混合有意义吗？在这个主题中，我们的目标是形成和证明沿子序列的(强)混合的一个版本，以及这样一个沿子序列的Wiener引理的适当版本。建议的第二部分的一个(较长期的)目的是为具有无限均值(不一定是规则变化的)的更新序列提供Erdos Feller Pollard定理的一个分析证明(现有证明使用概率方法)。

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).