Generalized spectral decomposition of the β-adic baker's transformation and intrinsic irreversibility

We construct a generalized spectral decomposition of the Frobenius-Perron operator of the β-adic baker's transformation using a general iterative operator method applicable in principle for any mixing dynamical system. The eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one. We explicitly define appropriate generalized function spaces, which provide mathematical meaning to the formally obtained spectral decomposition. The unitary Frobenius-Perron evolution of densities, when extended to the generalized function spaces, splits into two semigroups, one decaying in the future and the other in the past. This split, which reflects the asymptotic evolution of the forward and backward K-partitions, shows the instrinsic irreversibility of the baker's transformation.