Entropy on regular trees

TL;DR

This paper investigates the properties of topological entropy in tree shifts, demonstrating its infimum nature, extending classical results, and providing new methods for approximation and analysis on regular trees.

Contribution

It establishes that tree shift entropy equals its infimum, extends entropy dominance to arbitrary subshifts, and introduces an efficient numerical method for entropy approximation on regular trees.

Findings

Tree shift entropy equals its infimum, simplifying calculations.

Entropy of the hard square tree shift increases with the tree degree k.

Strip entropy approximations converge to the golden mean tree shift entropy for k=2 to 8.

Abstract

We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is $\mathbb Z$. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on $\mathbb Z^2$, we show that the entropy of the hard square tree shift on the regular $k$-tree increases with $k$, in contrast to the case of $\mathbb Z^k$. We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for $k=2,\dots,8$ and propose that this holds for all $k \geq 2$. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the $k$-tree for all $k$. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.