2021/06/22 by Chiba, Hayato, Ikeda, Masahiro, Ishikawa, Isao
#Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2106.12087
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space H with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace X of H and its dual space X'. The three topological spaces X ⊂ H ⊂ X' is 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 types (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ration 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.