We introduce and characterise time operators for unilateral shifts and exact endomorphisms. The associated shift representation of evolution is related to the spectral representation by a generalized Fourier transform. We illustrate the results for a simple exact system, namely the Renyi map.

We construct a spectral decomposition of the Frobenius-Perron operator for the Chebyshev polynomials of the first kind. We defined a suitable dual pair or rigged Hilbert space which provides mathematical meaning to the spectral decomposition. The spectra of the even Chebyshev maps do not contain...