Mixing for continuous time dynamical systems, and invariance principles for time-one maps

在最简单的混沌系统中，“记忆丧失”或相关性衰减通常以指数速度发生。此外，相关性衰减的速率是稳定的（不受底层系统的微小变化的影响），这对应用程序很重要。然而，对于连续时间系统（流），还没有证明相关的稳定指数衰减的例子。然而，最近的合作已经导致稳定的衰减率是超多项式（比任何多项式速率更快）的大型流动。这一提议的目的之一是增加我们对衰减率及其流动稳定性的理解。我们打算考虑的例子包括著名的洛伦兹吸引子。在一个相关的方向上，某些气体模型（平面周期性洛伦兹流）被引入作为布朗运动的模型。一个被称为几乎确定不变性原理（ASIP）的统计定律精确地将其与布朗运动联系起来。最近的工作建立了洛伦兹流的ASIP。在实际应用中，通常以固定的时间间隔进行测量（对应于流的时间- 1图）。结果表明，时间一图的ASIP比流动本身的ASIP更困难，因此，本项目的一个重要目标是建立平面周期性洛伦兹流动的时间一图的ASIP。

In the simplest chaotic systems, ``loss of memory'' or decay of correlations often occurs exponentially quickly. Moreover, the rate of decay of correlations is stable (unaffected by small changes to the underlying system) which is important for applications. However, for continuous time systems (flows) there are no examples where stable exponential decay of correlations has been proved. Nevertheless, recent collaborations have lead to stable decay rates that are superpolynomial (faster than any polynomial rate) for large classes of flows. One of the aims of this proposal is to increase our understanding of decay rates and their stability for flows. The examples that we aim to consider include the well-known Lorenz attractor. In a related direction, certain gas models (planar periodic Lorentz flows) were introduced as models for Brownian motion. A statistical law called the almost sure invariance principle (ASIP) makes precise the connection with Brownian motion. Recent work established the ASIP for the Lorentz flow. In applications, measurements are usually taken at regular intervals of time (corresponding to the time-one map of the flow). It turns out that the ASIP for the time-one map is more difficult than the ASIP for the flow itself, and hence an important aim of this project is to establish the ASIP for the time-one map of the planar periodic Lorentz flow.