Generalized spectral decompositions of mixing dynamical systems

Abstract

Abstract We introduce a method for the explicit computation of the eigenvalue problem of the evolution operator of mixing dynamical systems. The method is based on the subdynamics decomposition of the Brussels–Austin groups directed by Professor I. Prigogine. We apply the method to three different representatives of mixing systems, namely, the Renyi maps, baker's transformations, and the Friedrichs model. The obtained spectral decompositions acquire meaning in suitable rigged Hilbert spaces that we construct explicitly for the three models. The resulting spectral decompositions show explicitly the intrinsic irreversibility of baker's transformations and Friedrichs model and the intrinsically probabilistic characters of the Renyi maps and baker's transformations. The dynamical properties are reflected in the spectrum because the eigenvalues are the powers of the Lyapunov times for the Renyi and baker systems and include the lifetimes for the Friedrichs model. © 1993 John Wiley & Sons, Inc.