Spectral Theory of Dynamical Systems

Abstract

We will introduce now the notions of correlation functions and their Fourier transforms which play an important role in various applications of the theory of dynamical systems. Suppose {T ^t} is a flow on a measure space (M,M,μ) and f ∈ L ² (M,M,μ). By the correlation function corresponding to f we mean the function b _f (t) = ∫ _M f(T ^t x)f(x)dμ A number of statistical properties of the dynamical system may be characterized by the limit behavior of the differences [b _f (t) — (∫ _M f dμ)²]- In the case of mixing, these expressions tend to zero as t → ∞. If the convergence is fast enough one may write the above expressions in the form

$${b_f}\left( t \right) - {\left( {\int_M {fd\mu } } \right)^2} = \int_{ - \infty }^\infty {\exp \left( {i\lambda t} \right) \cdot {\rho _f}\left( \lambda \right)d\lambda } $$

. The function ρ _f (λ) in this representation is called the spectral density of. The set of those λ for which ρ _f (λ) is essentially non-zero for typical ∫, characterizes in some sense the frequencies playing the crucial role in the dynamics of the system under consideration. It is sometimes said that the system “produces a noise” on this set. The investigation of the behavior of the functions ρ _f (t), as well as of their analogs ρ _f (n) n ∈ Z, in the case of discrete time, is very important both for theory and for applications.