Explicit entropy bounds for symmetric nearest-neighbor subshifts

TL;DR

This paper introduces a combinatorial approach to compute the topological entropy of symmetric nearest-neighbor subshifts, providing explicit bounds and convergence rates in arbitrary dimensions.

Contribution

It offers a new combinatorial method with explicit entropy bounds and convergence rates, simplifying previous algebraic techniques for symmetric nearest-neighbor subshifts.

Findings

Provides explicit entropy bounds involving pattern counts C_n.

Establishes a convergence rate for entropy approximation in any dimension.

Simplifies the proof of Friedland's result with an elementary combinatorial approach.

Abstract

We provide another approach to Friedland's result that the topological entropy $h$ of a symmetric nearest-neighbor subshift is computable. Instead of the previous algebraic technique, our approach is mostly combinatorial and involves only counts of locally admissible patterns $C_n$ of a cube $[1,n]^d$ in $\mathbb Z^d$. The main idea is a reflection-gluing construction: we flip admissible patterns and merge them along their boundaries. In addition to a short and elementary proof, another advantage is that our approach yields an explicit convergence rate in arbitrary dimensions, whereas obtaining such a rate is already complicated for $\mathbb Z^3$ in Friedland's approach. In particular, we show that for every $n\ge 1$, \[ \frac{1}{n^d}(\log C_{n+1} - q_d(n)\log|\Sigma|) \le h \le \frac{1}{n^d} \log C_n, \] where $\Sigma$ is the alphabet and \[ q_d(n)=(2^d-1)\sum_{k=0}^{d-1} \frac{\binom{d}{k}}{2^d-2^k}\, n^k. \]