Correlation of dynamical observables

For a probability-preserving transformation $T$ and $f,h\in L^2$ , their correlation at time $n\geq0$ is

c_n(f,h)=\langle U_T^nf,h\rangle =\int_X f(T^nx)\overline{h(x)}\,d\mu(x).

For invertible $T$ , negative times are also defined. The centered correlation subtracts $(\int f\,d\mu)\overline{\int h\,d\mu}$ ; the autocorrelation takes $h=f$ .

Events and representations

For $f=1_A,h=1_B$ , correlation equals $\mu(T^{-n}A\cap B)$ . It is a matrix coefficient of the Koopman representation, with the sign of $n$ adjusted for the inverse-pullback convention.