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On the entropies of subshifts of finite type on countable amenable groups

Sebastián Barbieri

TL;DR

The paper develops a framework to transfer entropy information between subshifts of finite type on countable amenable groups via group charts and a cocycle-based embedding, establishing an entropy addition formula in the free-chart regime. This leads to a mechanism by which the entropies of $H$-SFTs embed into $G$-SFT entropies up to an arbitrarily small additive constant, provided a translation-like action exists and $H$ is suitably finitely presented. In the principal case where $H=oldsymbol{bZ}^2$, the authors (subject to a corrigendum) relate $ ext{E}_{ ext{SFT}}(G)$ to $ ext{E}_{ ext{SFT}}(oldsymbol{bZ}^2)$ for a broad class of finitely generated amenable groups with decidable word problem, implying the entropy set is the non-negative upper semi-computable numbers in many instances. The results yield complete entropy characterizations for several group families (polycyclic-by-finite, products, countable amenable, branch groups) and unify entropy realization across these classes, highlighting both the potential and current limits of the method, notably in light of the corrigendum concerning Theorem 4.7. Overall, the work provides a powerful mechanism to describe which entropy values can occur for SFTs on large families of groups, with implications for computability and group-theoretic structure.

Abstract

Let $G,H$ be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary $H$-subshift into a $G$-subshift. Using an entropy addition formula derived from this formalism we prove that whenever $H$ is finitely presented and admits a subshift of finite type (SFT) on which $H$ acts freely, then the set of real numbers attained as topological entropies of $H$-SFTs is contained in the set of topological entropies of $G$-SFTs modulo an arbitrarily small additive constant for any finitely generated group $G$ which admits a translation-like action of $H$. In particular, we show that the set of topological entropies of $G$-SFTs on any such group which has decidable word problem and admits a translation-like action of $\mathbb{Z}^2$ coincides with the set of non-negative upper semi-computable real numbers. We use this result to give a complete characterization of the entropies of SFTs in several classes of groups. Corrigendum: An error has been found in the proof of Theorem 4.7. We have added a corrigendum appendix which explains the error, discusses possible solutions and details which results from Section 5 still hold (the only result that is no longer proven is Corollary 5.12). We also provide an update on the state of the art concerning the questions asked in Section 6.

On the entropies of subshifts of finite type on countable amenable groups

TL;DR

The paper develops a framework to transfer entropy information between subshifts of finite type on countable amenable groups via group charts and a cocycle-based embedding, establishing an entropy addition formula in the free-chart regime. This leads to a mechanism by which the entropies of -SFTs embed into -SFT entropies up to an arbitrarily small additive constant, provided a translation-like action exists and is suitably finitely presented. In the principal case where , the authors (subject to a corrigendum) relate to for a broad class of finitely generated amenable groups with decidable word problem, implying the entropy set is the non-negative upper semi-computable numbers in many instances. The results yield complete entropy characterizations for several group families (polycyclic-by-finite, products, countable amenable, branch groups) and unify entropy realization across these classes, highlighting both the potential and current limits of the method, notably in light of the corrigendum concerning Theorem 4.7. Overall, the work provides a powerful mechanism to describe which entropy values can occur for SFTs on large families of groups, with implications for computability and group-theoretic structure.

Abstract

Let be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary -subshift into a -subshift. Using an entropy addition formula derived from this formalism we prove that whenever is finitely presented and admits a subshift of finite type (SFT) on which acts freely, then the set of real numbers attained as topological entropies of -SFTs is contained in the set of topological entropies of -SFTs modulo an arbitrarily small additive constant for any finitely generated group which admits a translation-like action of . In particular, we show that the set of topological entropies of -SFTs on any such group which has decidable word problem and admits a translation-like action of coincides with the set of non-negative upper semi-computable real numbers. We use this result to give a complete characterization of the entropies of SFTs in several classes of groups. Corrigendum: An error has been found in the proof of Theorem 4.7. We have added a corrigendum appendix which explains the error, discusses possible solutions and details which results from Section 5 still hold (the only result that is no longer proven is Corollary 5.12). We also provide an update on the state of the art concerning the questions asked in Section 6.

Paper Structure

This paper contains 20 sections, 31 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Shift spaces
  4. Topological entropy
  5. Realization of entropies of subshifts of finite type
  6. Group charts and the addition formula
  7. Reducing the entropy of a chart
  8. Conditions for the existence of free charts
  9. Characterization of entropies: the case $H = \mathbb{Z}^2$
  10. Consequences
  11. Polycyclic-by-finite groups
  12. Products of infinite finitely generated groups
  13. Countably infinite amenable groups
  14. Branch groups
  15. Final remarks
  16. ...and 5 more sections

Key Result

Theorem \ref{theorem_HG}

Let $G,H$ be finitely generated amenable groups and let $\mathcal{E}_{\text{SFT}}(H)$ and $\mathcal{E}_{\text{SFT}}(G)$ respectively denote the set of real numbers attainable as topological entropies of an SFT in each group. Suppose that Then, for every $\varepsilon >0$ there exists a $G$-SFT $X$ such that $h_{top}(G\curvearrowright X) < \varepsilon$ and

Figures (4)

  • Figure 1: The circles $x,y,z$ represent points in the space $X$ while the arrows represent left multiplication by group elements. The cocycle equation states that the arrows commute: $\gamma(h_1h_2,x) = \gamma(h_1,y)\gamma(h_2,x)$.
  • Figure 2: The alphabet $\Sigma_{\texttt{snake}}$.
  • Figure 3: On the left we see a local patch of $X_{\texttt{snake}}$. The value of the $\mathbb{Z}$-cocycle $\gamma_{\texttt{snake}}(n,x)$ corresponds to the vector of $\mathbb{Z}^2$ obtained by following the arrow at the origin $n$ times. On the right we see a local patch of a configuration of $X^{\texttt{free}}_{\texttt{snake}}$. As cycles are forbidden, the cocycle induces a free action.
  • Figure 4: The subshift $Y_{\gamma}[X]$ is obtained by "overlaying" the copies of $H$ induced by $\gamma$ on $X$ with configurations of $Y$.

Theorems & Definitions (81)

  • Theorem \ref{theorem_HG}
  • Theorem \ref{theorem_caract_entropies_G_z2_translation_like}
  • Remark \ref{theorem_caract_entropies_G_z2_translation_like}
  • Definition \ref{theorem_caract_entropies_G_z2_translation_like}
  • Definition \ref{theorem_caract_entropies_G_z2_translation_like}
  • Remark \ref{theorem_caract_entropies_G_z2_translation_like}
  • Theorem \ref{theorem_caract_entropies_G_z2_translation_like}: Lind Lind1984
  • Definition \ref{theorem_caract_entropies_G_z2_translation_like}
  • Theorem \ref{theorem_caract_entropies_G_z2_translation_like}: Hochman and Meyerovitch HochmanMeyerovitch2010
  • Definition \ref{theorem_caract_entropies_G_z2_translation_like}
  • ...and 71 more