Some Developments on Noisy Dynamical Systems

In this book we introduce some developments in one of the modern and important branches of mathematics which is called Noise Dynamical System or also called Random Dynamical System. First, we make some development in metric dynamical system and study some important concepts such as mixing, weakly mixing, ergodicity, transitivity and exactness metric dynamical system. And then we study the random dynamical systems (RDS's) and show how can we generate RDS from another RDS (or two RDS's). Furthermore, we state the definition of the random fixed point and random periodic point in RDS and proved some new results related with these two concepts. And then go to the concepts of random sets in RDS's and study some new properties of such sets and defined new kinds of these sets in RDS's such as minimal random set, transitive random set, thin random set, and small random set, which are used later in constructing and studying new kinds of RDS's such as minimal RDS, transitive RDS, thin RDS and strong RDS. And we prove important properties related with these systems.