Tree-Shifts: The entropy of tree-shifts of finite type

TL;DR

This paper investigates the entropy of tree-shifts of finite type, showing its computation relates to nonlinear recurrence equations and revealing rich phenomena including multinacci numbers, with analysis of boundary condition effects.

Contribution

It establishes the equivalence between entropy calculation and nonlinear recurrence solutions and explores boundary condition impacts on entropy in tree-shifts.

Findings

Entropy of binary Markov tree-shifts is either 0 or ln 2.

Realization of multinacci numbers demonstrates rich phenomena.

Conditions for entropy coincidence under different boundary conditions.

Abstract

This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either $0$ or $\ln 2$. Meanwhile, the realization of a class of reals including multinacci numbers is elaborated, which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed.