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