Ruette, Sylvie Transitive topological Markov chains of given entropy and period with or without measure of maximal entropy. (English) Zbl 1439.37013 Pac. J. Math. 303, No. 1, 317-323 (2019). 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 Keywords:topological Markov chain; countable oriented graph; topological entropy PDF BibTeX XML Cite \textit{S. Ruette}, Pac. J. Math. 303, No. 1, 317--323 (2019; Zbl 1439.37013) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.