Ergodic Theory of Weakly Hyperbolic Dynamical Systems

德米特里•Dolgopyat。弱双曲动力系统的遍历理论。光滑遍历理论研究可由有限个数参数描述的自治系统的长时间渐近问题。一个令人惊奇的事实是，即使这样的系统的演化是完全确定的敏感依赖于初始条件（即附近轨迹的指数散度），但在存在渐近测量（平衡状态）的意义上，往往导致随机行为，使得大多数轨道根据该测量在相空间中分布。在过去的几十年中，已经发展了证明渐近测度存在性的有力方法。这个理论已经在纯数学的许多领域得到了惊人的应用，最著名的是数论和李论。它是否能应用于现实生活问题还不太清楚，因为在实践中，系统可以用有限数量的参数来描述的假设仅仅是一种理想化，在短期内是无害的，但从长远来看可能是至关重要的。该项目的目的是结合非均匀双曲和部分系统领域的最新进展，以开发有效和可靠的工具来解决以下问题。(1)确定收敛到平衡状态的速度；(2)描述系统在各种扰动下平衡态的变化。首先，本项目发展的理论是微扰性质的，描述了熟知的系统的小微扰，特别是具有紧对称群的横向双曲系统，但最终它应该适用于一类大的动力系统。这项研究对任何使用常微分方程的科学领域都有意义。

Dmitry Dolgopyat.Ergodic theory of weakly hyperbolic dynamical systems.Smooth ergodic theory deals with long time asymptotic of autonomous systems which can be described by finite number of parameters. An amazing fact is that even though the evolution of such systems is completely deterministic sensitive dependence on initial conditions (that is exponential divergence of nearby trajectories)often leads the random behavior in the sense that there is asymptoticmeasure (equilibrium state) such that most orbits become distributed in the phase space according to this measure. During past decades powerful methods for proving the existence of the asymptotic measure have been developed. This theory already has found spectacular applications in many areas of pure mathematics most notably number theory and Lie theory. It is less clear if it can be appliedto real life problems because in practise the assumption that the system can be described by a finite number of parameters is merely an idealization which is harmless in short time range but may be crucial in the long run. The aim of this project is to combine recent advances in the fields of non-uniformly hyperbolic and partially systems in order to develop effective and reliable toolsfor solving the following problems. (1) Determining the rate of convergence to equilibrium;(2) Describing the change of equilibrium under various perturbations of the system.At first the theory developed in this project would be of perturbative nature describing small perturbations of well understood systems in particular of transversely hyperbolic systems with compact symmetry group but eventually it should be applicable to a large class of dynamical systems. This research will be of interest in any field of science where ordinary differential equations are used.