Characterization of when expanding toral endomorphisms admit linear Markov partitions in all dimensions

Establish that for every expanding n×n integer matrix A, the induced toral endomorphism f_A: T^n → T^n admits a linear Markov partition if and only if there exists k ∈ N such that A^k is diagonalizable with integer eigenvalues.

Background

The paper proves that if some power of an expanding integer matrix A is diagonalizable with integer eigenvalues, then the induced expanding toral endomorphism f_A admits a linear Markov partition in any dimension. In dimension 2, the authors further show a complete characterization: f_A admits a smooth (linear) Markov partition if and only if some power of A is diagonalizable with integer eigenvalues.

Motivated by Cawley’s classification for hyperbolic toral automorphisms, the authors conjecture that this 2-dimensional necessity-and-sufficiency criterion extends to all dimensions for expanding toral endomorphisms.

References

There is an obvious conjectured generalization of this result, mirroring Theorem \ref{thm:Cawley}, which we leave for future work. Let $A$ be an expanding $n\times n$ integer matrix. Then the induced toral endomorphism $f$ admits a smooth, in fact linear, Markov partition if and only if for some $k\in N$, $Ak$ is diagonalizable with integer eigenvalues.

— Smoothness of Markov Partitions for Expanding Toral Endomorphisms  (2604.00257 - Hughes et al., 31 Mar 2026) in Introduction (Section 1), preceding and including the unnumbered Conjecture