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Entropy conjugacy for Markov multi-maps of the interval. (English) Zbl 1471.37044

The authors consider a class \(\mathcal{F}\) of Markov multi-maps of the unit interval. A multi-map of the unit interval is a set-valued function on \([0,1]\) that assigns to each point a closed subset of \([0,1]\). Each multi-map gives rise to a shift map on the space of its trajectories, which is a closed and shift-invariant subset of \([0,1]^{\mathbb{Z}_+}\). Meanwhile, for each Markov multi-map \(F\) in \(\mathcal{F}\), one may also associate to it a shift of finite types, which is induced by a 0-1 matrix \(M=M(F)\). The authors show that the two shift maps are (Borel) entropy conjugate to each other for every Markov multi-map in \(\mathcal{F}\). In particular, this implies that they have the same topological entropy and the same number of measures of maximal entropy. Furthermore, the authors identify all possible values of the topological entropies of the Markov multi-maps in \(\mathcal{F}\), which are exactly all positive rational multiples of the logarithms of the Perron numbers.
Reviewer: Peng Sun (Beijing)

MSC:

37E05 Dynamical systems involving maps of the interval
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B40 Topological entropy
28D20 Entropy and other invariants
15B34 Boolean and Hadamard matrices
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