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。这些动力系统模型中的许多都存在复杂的混沌行为。绝对连续不变量测度(ACIM)是研究离散动力系统模型混沌行为的有力工具。ACIM测量具有任意初始点的离散动力系统产生的混沌轨道上各点的渐近相对频率。动力系统的轨道在确定性意义上可以是非常复杂的，但在概率或统计意义上它可能不是混沌的。对于研究长期行为及其混沌性质，ACIM是一个非常有用的数学工具。我们怎么知道这样的ACIM存在？如果它存在，我们如何才能从分析和数字上找到acims？这些acims的特性是什么。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AIMS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，我计划在一维和更高维上研究一些混沌离散动力系统。首先，我将研究一族随机映射族(封闭系统)的无穷acims的存在性。此外，我们还将研究相应的具有空洞的开放动力系统的绝对连续条件不变测度和逃逸率。随机映射是一个离散时间动态系统，其中状态空间上的多个映射中的一个是根据固定概率或位置相关概率随机选择的，并应用于该过程的每次迭代中。随机映射在科学和工程的许多领域都有应用，如研究分形学、对量子力学中的干扰效应进行建模、计算度量熵以及预测金融市场。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、量子力学严格效应、数值计算、微分包含等科学与工程领域有着重要的应用。我们对绝对连续不变测度的存在性、逼近和性质感兴趣。第三，通过利用一维映射的当前和过去信息的过程，研究一维映射产生的二维混沌动力系统--具有记忆的映射的动力学。在许多实际情况下(如股票市场)，这些类型的二维动力系统是分析各种量的有用的数学模型。我们将研究有记忆的地图的SRB度量、AIMS和其他动力学。最后，研究几何马尔可夫更新过程(GMRP)的稳定性和控制性。GMRP是一个研究金融中期权价格的过程。

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.