Non-diagonalizability of the Frobenius-Perron operator and transition between decay modes of the time autocorrelation function

We show that for one-dimensional piecewise linear Markov maps the Frobenius-Perron operator evolving probability densities may admit Jordan blocks. Its spectral decomposition is obtained in that case using the formalism of the generalized master equation developed by MacKernan and Nicolis. For mixing piecewise linear Markov maps with two branches and a corresponding two-cell partition, it is shown that the particular situation occurring when the Frobenius-Perron operator restricted to piecewise linear functions is not diagonalizable is a transition between two different decay modes of the time autocorrelation function. The general case of an M-cell partition is also addressed.