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Transitive topological Markov chains of given entropy and period with or without measure of maximal entropy. (English) Zbl 1439.37013
Summary: We show that, for every positive real number \(h\) and every positive integer \(p\), there exist oriented graphs \(G, G'\) (with countably many vertices) that are strongly connected, of period \(p\), of Gurevich entropy \(h\), and such that \(G\) is positive recurrent (thus the topological Markov chain on \(G\) admits a measure of maximal entropy) and \(G'\) is transient (thus the topological Markov chain on \(G'\) admits no measure of maximal entropy).
MSC:
37B10 Symbolic dynamics
37B40 Topological entropy
37A30 Ergodic theorems, spectral theory, Markov operators
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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References:
[1] ; Dajani, Ergodic theory of numbers. Ergodic theory of numbers. Carus Mathematical Monographs, 29 (2002) · Zbl 1033.11040
[2] 10.1007/BFb0082364 · Zbl 0328.28008
[3] ; Gurevich, Dokl. Akad. Nauk SSSR, 187, 715 (1969)
[4] ; Gurevich, Dokl. Akad. Nauk SSSR, 192, 963 (1970)
[5] 10.2140/pjm.2003.209.365 · Zbl 1055.37020
[6] 10.2140/pjm.1988.134.325 · Zbl 0619.54031
[7] 10.1090/conm/135/1185102
[8] 10.1093/qmath/13.1.7 · Zbl 0104.11805
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