Computing the topological entropy of shifts

Abstract

Key words Shift dynamical systems, topological entropy, type-2 computability, labeled digraphs. MSC (2000) 37B10, 37B40, 03F60 Different characterizations of classes of shift dynamical systems via labeled digraphs, languages, and sets of forbidden words are investigated. The corresponding naming systems are analyzed according to reducibility and particularly with regard to the computability of the topological entropy relative to the presented naming systems. It turns out that all examined natural representations separate into two equivalence classes and that the topological entropy is not computable in general with respect to the defined natural representations. However, if a specific labeled digraph representation -namely primitive, right-resolving labeled digraphs -of some class of shifts is considered, namely the shifts having the specification property, then the topological entropy gets computable.

Computability and Dynamical Systems

“…This kind of result brings to the field of computability questions traditionally related to dynamical systems [29]. In particular, these include deriving necessary conditions for universality [18], computability of entropy [32], [54], [27], [26] [55], and understanding the "edge of chaos [35]." Also along this line, some work has been done concerning computability of dynamical systems seen from a statistical perspective [30], [21].…”

Computability of Topological Pressure for Shifts of Finite Type with Applications in Statistical Physics

“…In [13] it was shown that a corresponding machine does not exist computing the topological entropy. Since the topological entropy is the topological pressure for the null function, there is no such machine computing the topological pressure.…”

“…Next, Equation (2) gives T n v,u > 0 iff χ A * (X) (vu) = 1. Hence there is an algorithm computing the irreducible components of the transfer matrix, as shown in [13]. Furthermore, the characteristic polynomial of a matrix is computable with respect to a name of the matrix as input.…”

“…Since the topological pressure is a generalization of the topological entropy, as already mentioned, the following elaboration is a continuation of [13] where computability aspects of the topological entropy were considered. While in [13], it was possible to show the computability of the topological entropy for types of shift dynamical systems far beyond the shifts of finite type, the computability of the topological pressure is shown here only for shifts of finite type. However, even for shifts of finite type, the concept is applicable to a wide class of so called one dimensional spin systems, mainly investigated in theoretical statistical physics.…”

Beta-Shifts, Their Languages, and Computability

“…For symbolic dynamical systems more general than the β-shifts, progress has been made recently in finding good classifications of what computational properties of their languages are needed to ensure computability of topological invariants, especially the topological entropy [32,35,14,13]. However, for β-shifts, not much is known when the shifts are no longer assumed to be sofic, bar some very interesting, but little known, results by Johnson [17].…”