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.