Spectral Representations of the Bernoulli Map

We discuss spectral representations of the Perron-Frobenius operator, U , associated with a highly chaotic map. The continuous spectrum of U does not contain (except coincidentally) information about physically accessible quantities such as decay rates of correlation functions. We show constructively that decay rates can be incorporated into a generalized spectral decomposition of U if its domain is restricted to smooth functions. 1. Introduction Several recent papers [1][2][3][4][5] discuss generalized spectral representations of the Perron-Frobenius operator, U , for simple highly chaotic maps. This operator governs the time evolution of probability densities, which are often taken to belong to the Hilbert space L 2 of square integrable functions over phase space. For mixing systems the spectrum of U operating on the Hilbert space is continuous, and in some simple cases one can construct an explicit spectral representation of U as an integral over the continuous spectrum. If its ..