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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 arestrongly connected, of period p, of Gurevich entropy h, such that G is positiverecurrent (thus the topological Markov chain on G admits a measure of maximalentropy) and G' is transient (thus the topological Markov chain on G' admits nomeasure of maximal entropy).

Topological Markov chains of given entropy and period with or without measure of maximal entropy