Time-Correlation Functions of One-Dimensional Transformations

Broken line transformations consist of three elementary processes; the staircase processes, the rotation around a fixed point and the hopping between different branches. Their relative weights determine the main feature of the time-correlation function <xn || x0> of mapping xn = f(xn−1), (n = 1, 2, ⋯). It is shown that , where is a linear operator closely related to the Frobenius-Perron operator. Eigenvalues and eigenfunctions of determine the time-correlation function. By means of this method the time-correlation functions of typical broken line transformations are calculated exactly and their features are clarified in terms of the three elementary processes and unstable periodic orbits.