Elementary cellular automata: spectral properties of Koopman operator for finite periodic realizations

In this paper we study Koopman operators associated with finite periodic realizations of elementary cellular automata. The spectra of these operators encompass various properties of the underlying dynamical systems, independently of any initial condition. A characterization of these spectra is proposed, leading to a spectral classification of all elementary cellular automata. This work is thought as a preliminary step for studying the spectra of Koopman operators for infinite realizations of elementary cellular automata.