A Numerical View of Spectral Decomposition Theorem for Sft

Finite type subshifts with finite number of symbols are important sort of discrete dynamical systems (often called symbolic dynamics) and are intensively studied [2, 3, 4, 5, 6]. Mañé proved a theorem (for finite type subshifts generated by 0, 1-square matrices), stating that topological transitivity and topological mixing can be reduced to algebraic properties of associated matrices [5]. In the paper [3] algorithmical conditions for topological transitivity and mixing were considered. The Spectral Decomposition Theorem is an important result in the theory of dynamical systems, giving a separation of a nonwandering set into basic sets with topological transitivity and elementary sets with topological mixing [1]. In this paper an algorithmizable method of Spectral Decomposition Theorem [1] for subshfits of finite type (SFT) is formulated. The classical Smale’s Spectral Decomposition Theorem [4, 5, 6] and algorithmic results contained in papers [2, 3] has been utilised.