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。首先，我将研究一组随机映射（封闭系统）的无穷acim的存在性。此外，我们将研究具有空穴的开放动力系统的绝对连续条件不变测度和逃逸率。随机映射是一种离散时间动态系统，它根据固定概率或位置依赖概率在状态空间上随机选择一个映射，并应用于过程的每次迭代。随机映射在科学和工程的许多领域都有应用，比如分形的研究、量子力学中的干扰效应建模、度量熵的计算和金融市场的预测。其次，我将研究多值映射的动力学。多值映射在混沌同步、经济学、量子力学中的严格效应、数值和微分内含物等许多科学和工程领域发挥着重要作用。我们对绝对连续不变测度的存在性、近似性和性质感兴趣。第三，我将研究带记忆地图的动力学，它是由一维地图通过利用一维地图的当前和过去信息的过程产生的二维混沌动力系统。有许多实际情况（如股票市场），这些类型的二维动力系统是分析各种数量的有用数学模型。我们将研究SRB测量，acims和其他具有记忆的地图动态。最后，我将研究几何马尔可夫更新过程（GMRP）的稳定性和控制。GMRP是研究金融领域期权价格的一种方法。

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. Firstly, I will study the existence of infinite acims for a family of random maps (closed systems). Moreover, we will study absolutely continuous conditional invariant measures and escape rates of the corresponding open dynamical systems with holes. A random map is a discrete time dynamical system, where one of a number of maps on the state space is selected randomly according to fixed probabilities or position dependent probabilities and applied in each iteration of the process. Random maps have applications in many areas of science and engineering such as in the study of fractals, in modelling interference effects in quantum mechanics, in computing metric entropy, and in forecasting the financial markets. Secondly, I will study dynamics of multi-valued maps. Multi-valued maps play an important role in many area of Science and Engineering such as in chaos synchronization, economics, rigorous effects in quantum mechanics, numerics and differential inclusions. We are interested in existence, approximations and properties of absolutely continuous invariant measures. Thirdly, I will study dynamics of maps with memory which are two dimensional chaotic dynamical systems generated by one dimensional map via a process which uses current and past information of the one dimensional map. There are many practical situations (such as stock market) where these type of two dimensional dynamical systems are useful mathematical models for analyzing various quantities. We will study SRB measures, acims and other dynamics of maps with memory. Finally, I will study the stability and control of the Geometric Markov Renewal Processes (GMRP). A GMRP is a process for the study of option prices in finance.