Non-existence of a universal zero-entropy system
The author studies the problem of the existence of universal dynamical systems with zero entropy. A topological dynamical system \((X,T)\) is universal in a class \(\mathcal S\) of measure-theoretic dynamical systems when \((X,\mu,T)\) belongs to \(\mathcal S\) for any \(T\)-invariant measure \(\mu\) and if for a given measure-theoretic system in \(\mathcal S\) there exists a measure \(T\)-invariant \(\mu\) such that \((X,\mu,T)\) is measure-theoretic isomorphic to that given system. The main result of this reviewed article states that there does not exist a topological dynamical system which is universal in the class of automorphisms with zero entropy whose topological entropy is equal to zero. Important ingredients for the proof are the notions of measure-theoretic complexity and the symbolic complexity of the system, and when the complexity is dominated by a sequence of numbers. One of the key arguments for the proof (Lemma 2) is about the existence of dynamical systems with zero entropy with the complexity dominated for a given non-decreasing sequence of numbers.
scientific article; zbMATH DE number 6176549