Perron–Frobenius degrees of Perron numbers: which data control the minimal dimension of a non-negative integral matrix realization (Yazdi, Questions 5.1–5.4 and Remark 3.9)

By Lind's converse to the integer Perron–Frobenius theorem, a real number is the spectral radius of a primitive non-negative integral matrix exactly when it is a Perron number — a real algebraic integer greater than 1 whose other Galois conjugates are all smaller in modulus. The Perron–Frobenius degree $d_{PF}(\lambda)$, the least dimension of such a matrix, and its irreducible analogue $d_{PF}^{irr}(\lambda)$ measure the smallest finite-state (shift-of-finite-type) realization of the entropy $\log\lambda$. Yazdi proved the first explicit upper bounds, of the form $(8d/(1-\rho))^{d^2}\tau_{\min}(\mathcal{O}_K)^{d/2}$ and $(8d/(1-\rho))^{d^2}\sqrt{D_K}$, where $d$ is the algebraic degree, $\rho$ the spectral ratio, and $\tau_{\min}$, $D_K$ are arithmetic invariants of the number field $K=\mathbb{Q}(\lambda)$. The paper's Section 5 asks which data are truly necessary. The open questions are: (1) can $d_{PF}(\lambda)$ be bounded in terms of $d_{PF}^{irr}(\lambda)$, and does equality always hold; (2) does a uniform bound $B(d,\rho)$ free of arithmetic field data exist (the author conjectures not); (3) how do the bounds improve for totally real Perron, Pisot, Salem and bi-Perron numbers; (4) for Pisot numbers, does a bound depending only on the degree exist; and (5) can Thurston's variant of Lind's construction always be taken primitive.

Background

A square matrix with non-negative integer entries has a real non-negative eigenvalue of maximal modulus, its spectral radius (Perron–Frobenius theorem). Such a matrix is primitive if some power of it has all entries strictly positive, and irreducible if it is not permutation-similar to a block upper-triangular matrix — equivalently, the directed graph with an edge $i\to j$ whenever the $(i,j)$ entry is positive is strongly connected; every primitive matrix is irreducible. A Perron number is a real algebraic integer $\lambda>1$ all of whose other Galois conjugates have modulus strictly less than $\lambda$. Lind's converse to the integer Perron–Frobenius theorem states that the spectral radii of primitive integral non-negative matrices are exactly the Perron numbers; equivalently, the numbers $e^h$ with $h$ the topological entropy of a mixing shift of finite type — a shift-invariant closed set of bi-infinite sequences over a finite alphabet determined by finitely many forbidden blocks, also called a topological Markov shift, with the matrix as adjacency matrix. Lind also gave a method for producing all integral primitive matrices with a given Perron number as spectral radius.

For a Perron number $\lambda$, the Perron–Frobenius degree $d_{PF}(\lambda)$ is the least dimension of an integral primitive matrix with spectral radius $\lambda$, and the irreducible Perron–Frobenius degree $d_{PF}^{irr}(\lambda)$ is the same minimum over irreducible matrices; since primitive matrices are irreducible, $d_{PF}^{irr}(\lambda)\le d_{PF}(\lambda)$. These numbers measure the minimal number of states of a finite-state dynamical system realizing the entropy $\log\lambda$. One always has $d_{PF}(\lambda)\ge d$, the algebraic degree, and the inequality can be strict (Lind). The inverse spectrum problem — which multisets of nonzero complex numbers occur as the nonzero spectra of primitive non-negative integer matrices — was solved by Boyle and Handelman and constructively by Kim, Ormes and Roush, and Boyle and Lind studied small polynomial matrix presentations; none of these results controls the dimension of the realizing matrix. Yazdi proved the first lower bounds for $d_{PF}(\lambda)$: using an idea of Lind, a lower bound in terms of the layout in the complex plane of the two largest Galois conjugates of $\lambda$, showing in particular that there are cubic Perron numbers with arbitrarily large Perron–Frobenius degree — a result previously known, unpublished, to Lind, McMullen and Thurston.

In the source paper, Yazdi proved the first explicit upper bounds. The spectral ratio of $\lambda$ is $\rho=\max_i|\lambda_i|/\lambda$, the maximum running over the conjugates $\lambda_i\ne\lambda$. For $\lambda$ of algebraic degree $d\ge3$, in the number field $K=\mathbb{Q}(\lambda)$ with ring of integers $\mathcal{O}_K$ (the subring of elements of $K$ that are algebraic integers), discriminant $D_K$, and minimal Hermite-like thickness $\tau_{\min}(\mathcal{O}_K)$ (a lattice invariant of $\mathcal{O}_K$ introduced by Bayer-Flückiger: essentially the minimum, over choices of inner product on $\mathbb{R}^d$, of the properly normalized square of the covering radius of the lattice $\mathcal{O}_K\subset\mathbb{R}^d$), Theorem 1.3 of the paper bounds $d_{PF}^{irr}(\lambda)$ above by each of $(8d/(1-\rho))^{d^2}\tau_{\min}(\mathcal{O}_K)^{d/2}$ and $(8d/(1-\rho))^{d^2}\sqrt{D_K}$. The proof passes through Bayer-Flückiger's upper bounds for Euclidean minima of number fields and Banaszczyk's transference theorems from the geometry of numbers, so arithmetic data of $K$ enter unavoidably. A companion result (Theorem 1.6) bounds $d_{PF}(\lambda)$ above by $\max\{2^{d^2}B(K,\rho),\kappa(K,\rho)\}$, still depending on $K$. The special classes named in the questions: a totally real Perron number is a Perron number all of whose Galois conjugates are real; a Pisot number is a real algebraic integer greater than 1 all of whose other conjugates lie strictly inside the unit circle; a Salem number is a real algebraic integer greater than 1 whose other conjugates lie in the closed unit disk with at least one on the unit circle; a unit (algebraic integer of norm $\pm1$) is bi-Perron if all conjugates other than $\lambda$ lie in the annulus $\lambda^{-1}<|z|<\lambda$, except possibly $\pm\lambda^{-1}$ — the class containing the stretch factors of pseudo-Anosov surface homeomorphisms (surface homeomorphisms of maximal dynamical complexity, which expand along one measured foliation and contract along another by a common factor, the stretch factor $\lambda$). Thurston, in 'Entropy in dimension one' (page 354), gave a new proof of Lind's converse whose output is an irreducible matrix, and assumed that it can be taken primitive. The questions the paper poses in its Section 5 — which data are truly necessary to control the degrees — are the present problem.

Problem Statement

Let $\lambda$ be a Perron number (a real algebraic integer $\lambda>1$ all of whose other Galois conjugates have modulus strictly less than $\lambda$). Let $d_{PF}(\lambda)$ denote the least dimension of an integral primitive non-negative matrix with spectral radius $\lambda$, and $d_{PF}^{irr}(\lambda)$ the least dimension of an integral irreducible such matrix; let $d$ be the algebraic degree of $\lambda$ and $\rho$ its spectral ratio $\rho=\max_i|\lambda_i|/\lambda$, the maximum running over the Galois conjugates $\lambda_i\ne\lambda$. Determine which natural data of $\lambda$ control these degrees, by resolving the following questions of Yazdi's Section 5 together with the primitivity issue of Remark 3.9:

(1) (Question 5.1) Are there upper bounds for $d_{PF}(\lambda)$ in terms of $d_{PF}^{irr}(\lambda)$? Does $d_{PF}(\lambda)=d_{PF}^{irr}(\lambda)$ always hold?

(2) (Question 5.2) Is there an upper bound $B(d,\rho)$, depending only on the algebraic degree and the spectral ratio, for $d_{PF}^{irr}(\lambda)$ (respectively $d_{PF}(\lambda)$), valid for arbitrary Perron numbers $\lambda$? The source expects a negative answer, meaning that in the main theorem the arithmetic information on the ring of integers $\mathcal{O}_K$ cannot be overlooked.

(3) (Question 5.3) How can the general bound of the source's Theorem 1.3 be improved for special classes of algebraic integers, such as totally real Perron numbers, Pisot numbers, Salem numbers, or bi-Perron numbers? Here the general class of special algebraic integers is meant (the four listed classes being examples), and any strict improvement of the Theorem 1.3 bound counts.

(4) (Question 5.4) Is there an upper bound $B(d)$ depending only on the algebraic degree for $d_{PF}^{irr}(\lambda)$ (respectively $d_{PF}(\lambda)$), where $\lambda$ is an arbitrary Pisot number (a real algebraic integer greater than 1 all of whose other conjugates lie strictly inside the unit circle)?

(5) (Remark 3.9) Can the matrix produced by Thurston's variant of Lind's construction ('Entropy in dimension one', page 354) always be taken primitive, so that the irreducible bound of Theorem 1.3 upgrades directly to a bound on $d_{PF}(\lambda)$?

These five facets share one scientific objective — to identify exactly which invariants of a Perron number are the necessary and sufficient data for controlling the minimal dimension of its finite-state (non-negative integral matrix) realizations. A complete answer resolves all five; progress on individual facets is meaningful in proportion to how much of this objective it settles.

References

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