Explicit entropy bounds for symmetric nearest-neighbor subshifts

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 $n\ge 1$, $$\frac{1}{n^d}(\log C_{n+1}-q_d(n)\log \mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid })\le h\le \frac{1}{n^d}\log C_n,$$ where $\mathrm{\Sigma }$ is the alphabet and $$q_d(n)=(2^d-1)\sum _{k=0}^{d-1}\frac{\left(\genfrac{}{}{0px}{}{d}{k}\right)}{2^d-2^k}{n}^k\mathrm{.}$$