## The topological entropy of cellular automata is uncomputable.(English)Zbl 0770.58017

Let $$S$$ be a finite set, $$S^ \mathbb{Z}$$ the set of all functions $$\mathbb{Z} \to S$$ and $$\sigma: S^ \mathbb{Z} \to S^ \mathbb{Z}$$ the left shift, i.e. $$\sigma(x)_ i = x_{i+1}$$, for all $$i \in \mathbb{Z}$$. A cellular automaton is a continuous function $$S^ \mathbb{Z} \to S^ \mathbb{Z}$$ which commutes with $$\sigma$$. The authors show: (1) There is no algorithm which will take a description of a cellular automaton and determine whether it has zero topological entropy, or for any fixed $$\varepsilon > 0$$ compute its topological entropy to a tolerance $$\varepsilon$$. (2) An example of a cellular automaton with only one periodic point, having nontrivial non- wandering set and arbitrarily large topological entropy.
Reviewer: J.Ombach (Kraków)

### MSC:

 37B99 Topological dynamics 54C70 Entropy in general topology

### Keywords:

cellular automata; computability; topological entropy
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### References:

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