Dynamics of maps with memory, random maps, multi-valued maps and the geometric Markov Renewal processes

动力系统是科学、工程、经济、金融等领域许多问题的数学模型。许多动力系统模型都存在复杂的混沌行为。绝对连续不变测度（acim）是研究离散动力系统模型混沌行为的有力工具。一个acim测量由具有任意初始点的离散动力系统产生的混沌轨道各点的渐近相对频率。动力系统的轨道在确定性意义上可能非常复杂，但在概率或统计意义上却不一定是混沌的。acim是研究长期行为及其混沌性质的一个非常有用的数学工具。我们怎么知道存在这样一个星系呢？如果它存在，我们如何在分析和数值上找到它？这些原子的性质是什么？这些都是遍历理论和动力系统中有趣、重要和具有挑战性的问题。我的长期目标是通过acims和其他动力学为理论和计算方法的发展做出很大贡献。在接下来的5年里，我计划研究一些一维和高维的混沌离散动力系统。

Dynamical systems are mathematical models for many problems in science, engineering, economics, finance and other areas. Complicated chaotic behaviors occur for many of these dynamical system models. An absolutely continuous invariant measure (acim) is a powerful tool for the study of chaotic behavior of discrete dynamical system models. An acim measures asymptotic relative frequencies of points of chaotic orbits generated by a discrete dynamical system with any initial point. An orbit of a dynamical system can be very complicated in deterministic sense, however it may not be chaotic in probabilistic or statistical sense. An acim is a very useful mathematical tool for the study of long term behavior and their chaotic nature. How do we know that such an acim exists? If it exists, how can we find acims analytically and numerically? What are properties of these acims. These are interesting, important and challenging questions in Ergodic Theory and Dynamical Systems. My long term objective is to contribute largely for the development of theoretical and computational methods via acims and other dynamics. In the next 5 years, I plan to study a number of chaotic discrete dynamical systems in one and higher dimensions.