Little shift equivalence conjecture

Definitions and mathematical statement

Write $\mathbb Z_+=\{0,1,2,\ldots\}$. Two square matrices $A$ and $B$ over $\mathbb Z_+$, possibly of different sizes, are elementary strong shift equivalent if rectangular matrices $R,S$ over $\mathbb Z_+$ satisfy $A=RS$ and $B=SR$. They are strong shift equivalent over $\mathbb Z_+$ if there is a finite chain of square nonnegative integer matrices $A=A_0,A_1,\ldots,A_k=B$ with this property for every consecutive pair. Intermediate sizes and the length $k$ have no prescribed bound.

Let $A$ be an $r\times r$ nonnegative integer matrix whose only nonzero complex eigenvalue is $n$, of algebraic multiplicity one. Equivalently, \[ \det(tI-A)=t^{r-1}(t-n),\qquad n\in\mathbb Z_{>0}. \] The conjecture asserts that $A$ is strong shift equivalent over $\mathbb Z_+$ to the one-entry matrix $[n]$.

This formulation counts eigenvalues with algebraic multiplicity and allows nilpotent Jordan blocks at zero. In dynamical terms, construct the directed multigraph with $A_{ij}$ edges from vertex $i$ to vertex $j$, and shift its bi-infinite compatible edge paths. The requested equivalence says that this edge shift is topologically conjugate to the full shift on $n$ symbols. Agreement of characteristic polynomials alone is the hypothesis; exhibiting the required nonnegative factorizations, or proving that they always exist, is the additional content of the conjecture.

Short English statement

Does a nonnegative integer matrix with just one simple nonzero eigenvalue always present the corresponding full shift?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600001 (AMR-045-0001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 3.1. Original source indexed by AMR-045-0001. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.