Generalized Eigenvalues of the Perron-Frobenius Operators of Symbolic Dynamical Systems

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space \scrH with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace X of \scrH and its dual space X\prime. The three topological spaces X \subset \scrH \subset X\prime are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron-Frobenius operators of the one-sided and two-sided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on [0,1] modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron-Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.