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

TL;DR

This paper investigates the entropy of Markov tree-shifts defined by multiple transition matrices, providing a new method to characterize their complexity and compare different entropy definitions, along with analyzing invariant measures and topological properties.

Contribution

It introduces a novel approach to characterize the complexity function of Markov tree-shifts and compares various entropy definitions, addressing invariant measures and topological properties.

Findings

Derived a method to compute the complexity function for $k$-tree shifts.

Compared different entropy definitions and analyzed their properties.

Calculated entropies for specific examples and addressed invariant measures.

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.