Generalized eigenvalues of the Perron-Frobenius operators of symbolic dynamical systems

TL;DR

This paper develops a new approach to analyze the spectral properties of Perron-Frobenius operators in symbolic dynamical systems using generalized spectra and Gelfand triplets, providing insights into their asymptotic behavior.

Contribution

It introduces a novel algebraic construction of Gelfand triplets for symbolic dynamical systems and determines their generalized spectra, extending spectral analysis methods.

Findings

Generalized spectra of Perron-Frobenius operators are explicitly determined.

A new Gelfand triplet construction for symbolic systems is proposed.

Asymptotic formulas for operator iteration and convergence rates are provided.

Abstract

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace $X$ of $\mathcal{H}$ and its dual space $X'$. The three topological spaces $X \subset \mathcal{H} \subset X'$ 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.