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Entropy structure. (English) Zbl 1151.37020
The author proposes an entropy structure as a worthwhile invariant between topological dynamical systems which capture the relationship between entropy and resolution. Roughly speaking, given a topological dynamical system \((X,T)\) with \(X\) a compact metric space and \(T\) a homeomorphism, an entropy structure is any nondecreasing sequence of nonnegative functions on the simplex of \(T\)-invariant measures which converges to the entropy function \(h\) and which is equivalent in a natural way to a specific sequence reflecting the imperfection of uniformity in convergence.
The author shows that the entropy structure (as a class) is a topological invariant and that any two entropy structures of the same system are uniformly equivalent. The author also proves that a slight modification of Bowen’s entropy, and the entropy theories by Romagnoli, Brin-Katok, Orntein-Weiss and Newhouse can be obtained as entropy structures. Some failing candidates are also presented. The results are applied to the computation of symbolic extension entropy (without reference to zero-dimensional extensions). Some considerations on noninvertible continuous maps are also given.

MSC:
37B40 Topological entropy
28D20 Entropy and other invariants
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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