Diophantine Properties of Dynamical Systems: Quantitative, Connectivity and Proximality

该项目的中心主题是动态系统的丢番图和组合性质，特别是区间交换变换（IETs）。1993年，PI Boshernitzan开始了定量递归（QR）的研究，得到了著名的庞加莱递归定理的定量增强，声称动力系统的典型轨道（在某些温和条件下）返回任意接近其初始位置。定量增强解决了这种重复发生的速度和距离问题。事实证明，这些问题的答案是可能的根据空间的豪斯多夫维数。在过去的两年里，PI和他的学生Jon Chaika开始了一项研究，以定量的方式研究动力学中的连通性和邻近性现象。动力系统的连通性表现为，对于空间中典型的一对点，这些点的轨道闭包重合。连通性比递归性更具限制性。某些假设（如极小性或遍历性）被强加于这种现象。近性特性表现为一对典型的点是近的，即它们的轨道彼此任意接近。近邻现象并不是普遍存在的（但它适用于弱混合系统）。最近，PI和他的学生Chaika在联合工作中获得了定量连通性和邻近性（QCP）的一些初步结果，其中提出了一些重要的相关问题。这些结果推广和加强了丢番图分析（数论）中的一些经典结果，并与最近其他人在动力系统的“收缩目标”和“单调收缩目标性质”方面的工作有关。结果是详细和完整的，当应用于英语考试时很有趣。它是可逆的分段方向保持单位区间的等距，其研究的动机不仅是它们作为单位圆旋转的推广和保留勒贝格测度的最简单可逆映射的内在兴趣，而且是它们的应用，如紧致表面中的面积保持流和Teichmuller理论。QR的研究始于1993年，引起了国内外数学界的广泛关注，推动了该领域及相关领域的进一步研究和发展。QCP项目是QR项目的自然延续，希望它能对动力系统和相关领域的研究产生类似CR的影响。许多开放的问题和与其他数学领域的各种联系（遍历理论，IETs， Teichmuller理论，紧曲面上的保面积流）可能会让研究人员在未来几年里忙个不停。在过去的28年里，PI Boshernitzan一直负责挑选和指导莱斯队（以及所有感兴趣的学生）参加普特南数学竞赛。他指导过研究生，最近的一位是Jon Chaika(2010)，他自己也获得了数学界的广泛认可。

The central topic of this project is Diophantine and Combinatorial Properties of Dynamical Systems in general and Interval Exchange Transformations (IETs) in particular. In 1993 the PI, Boshernitzan, initiated study of Quantitative Recurrence (QR) obtaining a quantitative enhancement of the celebrated Poincare recurrence theorem claiming that typical orbits of dynamical systems (under some mild conditions) return arbitrarily close to their initial position. The quantitative enhancement addresses the question how soon and how close this recurrence takes place. It turns out that the answer to these questions is possible in terms of the Hausdorff dimension of the space. In the last two years the PI and his student, Jon Chaika, initiated research that studyies in a quantitative manner the Connectivity and Proximality phenomena in dynaThe Connectivity property of a dynamical system manifests itself in that, for a typical pair of points in the space, the closures of the orbits of these points coincide. Connectivity is more restrictive than Recurrence. Certain assumptions (like minimality or ergodicity) are imposed for the phenomenon to hold. The Proximality property manifests itself in that a typical pair of points is proximal, i.e. their orbits approach each other arbitrary close. The phenomenon of Proximality is not universal (but it holds for weakly-mixing systems). Some initial results on Quantitative Connectivity and Proximality (QCP) were obtained recently in joint work by the PI and his student, Chaika, where a number of important related questions were posed. These results generalize and strengthen some classical results in Diophantine analysis (in Number theory) and are related to the recent work of others in the area of "shrinking targets" and "monotone shrinking target property" of dynamical systems. The results are detailed and complete and of interest when applied to IETs. IETs are invertible piecewise orientation preserving isometries of the unit interval into itself and their study is motivated not only by their intrinsic interest as generalizations of the rotations of the unit circle and the simplest invertible maps which preserve Lebesgue measure, but also by their applications such as to area preserving flows in compact surfaces, and in Teichmuller theory. The study of QR initiated in 1993 attracted a lot of attention in the national and international mathematical community motivating much further research and development in this and related areas. The QCP project is a natural continuation of the QR project and it is hoped that it will have an impact on research in Dynamical Systems and related fields similar that of the CR. Many open problems and various connections with other fields of mathematics (Ergodic Theory, IETs, Teichmuller Theory, area preserving flows in compact surfaces), may keep researchers in the subject busy for years to come. Throughout the last 28 years, PI Boshernitzan has been responsible for the selection and coaching the Rice team (and all interested students) for the Putnam mathematical competition. He has mentored graduate students, the most recent one is Jon Chaika (2010) who himself has earned much recognition by the mathematical community.