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

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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.