Stochastic phenomena in deterministic dynamical systems: Introduction to statistical properties of dynamical systems

Abstract

In this paper, we introduce some probability results in the study of statistical properties of dynamical systems. With the evolution of a dynamical system $(M,f)$, a function $\phi$ defined on the phase space $M$ generates a process $\{\phi\circ f^n\}$. Even though it is given by a deterministic system, the process can exhibit many stochastic properties whenever the system is of certain chaotic behavior and the function $\phi$ is of some regularity. Such properties include the law of large number, the central limit theorem, the law of iterated logarithm, the principle of large deviations, local limit theorems, almost sure invariant principle, etc.