# Problem 116

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For a subshift $\Sigma _A$, $A$ on stable torus or stable $\mathbb{R}^n$ is an invariant. Describe it invariantly.

($\mathbb{R}^n = ?$, $A = ?$)

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### Here $\Sigma_A$ is presumably

Here $\Sigma_A$ is the shift of finite type defined by a square matrix $A$ with nonnegative integer entries. Let $A$ be
$k\times k$. Presumably stable $\mathbb R^n$'' is the subspace $V$ of $\mathbb R^k$ which is the eventual range of $A$, that is, the intersection of the images of $A^j$ over $j>0$, which equals the image of $A^k$. Restricted to $V$, $A$ is an invertible linear transformation, classified by (for example) the nonnilpotent part of the Jordan form of $A$. This data is a complete invariant of the shift equivalence class of $A$ over $\mathbb R$, hence an invariant of the shift equivalence class of $A$ over $\mathbb Z_+$, hence an invariant of topological conjugacy for the shift of finite type $\Sigma_A$.

Krieger's construction of the dimension module (dimension group of $\Sigma_A$ with automorphism induced by the shift) gives a descriptionwhich Bowen might have regarded as a more intrinsic or invariant'' description of the shift equivalence class of $A$ over $\mathbb Z_+$.

See Sections 7.4, 7.5 of [1] for more on this, with background and references.

### References

1. [LindMarcus1995] Lind D, Marcus B.  1995.  :xvi+495.