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 with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace of and its dual space . The three topological spaces 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 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.