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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.18164 |
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| _version_ | 1866917507894345728 |
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| author | Bui, Vuong |
| author_facet | Bui, Vuong |
| contents | 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 $n\ge 1$,
\[
\frac{1}{n^d}(\log C_{n+1} - q_d(n)\log|Σ|) \le h \le \frac{1}{n^d} \log C_n,
\]
where $Σ$ is the alphabet and
\[
q_d(n)=(2^d-1)\sum_{k=0}^{d-1} \frac{\binom{d}{k}}{2^d-2^k}\, n^k.
\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18164 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit entropy bounds for symmetric nearest-neighbor subshifts Bui, Vuong Dynamical Systems Combinatorics 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 $n\ge 1$, \[ \frac{1}{n^d}(\log C_{n+1} - q_d(n)\log|Σ|) \le h \le \frac{1}{n^d} \log C_n, \] where $Σ$ is the alphabet and \[ q_d(n)=(2^d-1)\sum_{k=0}^{d-1} \frac{\binom{d}{k}}{2^d-2^k}\, n^k. \] |
| title | Explicit entropy bounds for symmetric nearest-neighbor subshifts |
| topic | Dynamical Systems Combinatorics |
| url | https://arxiv.org/abs/2605.18164 |