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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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