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.
The topological pressure of dynamical systems theory is examined from a computability theoretic point of view. It is shown that for shift dynamical systems of finite type, the topological pressure is a computable function. This result is applied to a certain class of one dimensional spin systems in statistical physics. As a consequence, the specific free energy of these spin systems is computable. Finally, phase transitions of these systems are considered. It turns out that the critical temperature is not computable without further information on the system.
We prove that general topological dynamical systems on metric Cantor spaces can be embedded into cellular automata defined on Cayley graphs. Furthermore, we prove that it is possible to construct this embedding in such a way that topological entropy is preserved.
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.
Automata, Logic and Semantics
International audience
We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift
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