GENERALIZED MARKOV COARSE GRAINING AND SPECTRAL DECOMPOSITIONS OF CHAOTIC PIECEWISE-LINEAR MAPS

Spectral decompositions of the evolution operator for probability densities are obtained for the most general one-dimensional piecewise linear Markov maps and a large class of repellers. The eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal Markov partition equal the reciprocals of the zeros of the Ruelle zeta functions. The logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing. The space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit(s), so that local behavior of observables can be considered. The new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit(s). Finally, arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed.