ABSTRACT
In this chapter we review deterministic dynamical systems and their invariant measures. Deterministic dynamical systems are special cases of random dynamical systems, and theories of deterministic dynamical systems play an important role for the study of random dynamical systems. The existence and properties of absolutely continuous invariant measures for deterministic dynamical systems reflect their longtime behavior and play an important role in understanding their chaotic nature. The Frobenius-Perron operator for deterministic dynamical systems is one of the key tools for the study of invariant measures for deterministic dynamical systems. In Chapter 3 and Chapter 4 we will see that the Frobenius-Perron operator for random dynamical systems is a combination of the Frobenius-Perron operator of the individual component systems which are deterministic dynamical systems. In this chapter we focus our special attention on the class of piecewise monotonic and expanding deterministic dynamical systems. Moreover, we present stochastic perturbations of deterministic dynamical systems. For the Frobenius-Perron operator and existence of invariant measures we closely follow [2, 4, 9, 10] and the references therein. For the stochastic perturbations we closely follow [7, 8, 9, 11] and the references therein.
2.2 Deterministic dynamical systems
