Noncommutative entropy on generalized countable Markov shifts - Sun, Pingao (2025)
In this work, we study the relation between the noncommutative topological entropy of the canonical map for Exel-Laca algebra \\gamma_A:\\mathcal_A\\to\\mathcal_A and the Gurevich entropy of the respective Markov shift map \\sigma_A:\\Sigma_A\\to\\Sigma_A. We prove that these two entropies coincide for the renewal shift. In fact, one inequality holds for a more general situation whose proof follows from a series of classical results on entropy and an equality that relates the maps \\gamma_A and \\sigma_A; the proof of the other inequality is more technical and requires some specific properties of the renewal shift.