Spectral decomposition of the Renyi map

Abstract

The authors construct a generalized spectral decomposition of the Frobenius-Perron operator of the general beta -adic Renyi map using a general iterative operator method applicable in principle to any mixing dynamical system. They also explicitly define appropriate rigged Hilbert spaces, which provide mathematical meaning to the formally obtained spectral decomposition. The explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the Frobenius-Perron operator decreases as the smoothness of the domain increases. The reason for the change of the spectrum from the unit disk to isolated eigenvalues, is the existence of coherent states of the Frobenius-Perron operator, which are not infinitely differentiable.