Entropy for $k$-trees defined by $k$ transition matrices

Abstract

We study Markov tree-shifts given by $k$ transition matrices, one for each of its $k$ directions. We provide a method to characterize the complexity function for these tree-shifts, used to calculate the tree entropies defined by Ban and Chang arXiv:1509.08325 and Petersen and Salama arXiv:1712.02251. Moreover, we compare these definitions of entropy in order to determine some of their properties. The characterization of the complexity function provided is used to calculate the entropy of some examples. The question of existence of a specific type of invariant measures for such tree-shifts is addressed. Finally, we analyse some topological properties introduced by Ban and Chang arXiv:1509.01355 for the purpose of answering two of the questions raised by these authors.