[Paper Review] Some monoids of Pisot matrices

Artur Avila, Vincent Delecroix|arXiv (Cornell University)|Jun 11, 2015

semigroups and automata theory4 citations

TL;DR

This paper proves that certain monoids of non-negative integer matrices—specifically those arising from the fully subtractive and Brun continued fraction algorithms—are Pisot, meaning all eigenvalues except the dominant one have absolute value less than one. Using spectral bounds derived from the Perron-Frobenius eigenvector and matrix norms on invariant cones, the authors establish that primitive products of these matrices satisfy the Pisot property, with applications to exponential convergence in multidimensional continued fractions and dynamical systems with discrete spectrum.

ABSTRACT

We show that several monoids of non-negative integer matrices enjoy a Pisot property: each matrix in that monoid has only one eigenvalue with absolute value larger than one. These monoids come from multidimensional continued fractions, namely the fully subtractive and Brun continued fractions.

Motivation & Objective

• To establish that specific monoids of non-negative integer matrices arising from continued fraction algorithms are Pisot matrices.
• To prove that all primitive products of these matrices have spectral radii dominated by a simple eigenvalue with all other eigenvalues inside the unit disk.
• To provide a new, elementary proof technique based on matrix norms and invariant cones, differing from prior induction-based methods.
• To connect the Pisot property to dynamical systems and Lyapunov exponents, showing exponential convergence in continued fraction expansions.

Proposed method

• Uses the inequality $ \lambda_2 \leq \sup_{x \in v^\perp \setminus \{0\}} \frac{\|Ax\|}{\|x\|} $ to bound the second largest eigenvalue using the Perron-Frobenius eigenvector $ v $.
• Leverages the dynamics of the fully subtractive and Brun continued fraction algorithms to localize the dominant eigenvector within invariant cones $ D \subset \mathbb{P}(\mathbb{R}^d_+) $.
• Defines adapted sets $ D $ such that $ A^{(i)}D \subset D $, ensuring the existence of a uniform cone containing the dominant eigenspace for all matrix products.
• Applies the cocycle property $ A_{m+n}(x) = A_m(x) A_n(T^m x) $ to analyze long-term growth and spectral behavior of matrix products.
• Uses the log-integrability condition and Birkhoff's ergodic theorem to analyze Lyapunov exponents and derive $ \gamma_1 > 0 > \gamma_2 $, confirming the Pisot spectrum.
• Establishes that if a matrix product contains a specific subword (e.g., $ A_{Br}^{(3)} $), then $ \|A^3\| < 1 $, implying spectral dominance.

Experimental results

Research questions

• RQ1 Under what conditions do products of fully subtractive matrices form Pisot matrices?
• RQ2 How can the spectral properties of matrix monoids from continued fraction algorithms be characterized using geometric and dynamical constraints?
• RQ3 What role does the Perron-Frobenius eigenvector and its associated cone play in bounding the second eigenvalue?
• RQ4 Can the Pisot property be established without induction on characteristic polynomials, using norm-based spectral estimates?
• RQ5 How do Lyapunov exponents relate to the Pisot spectrum in the context of matrix cocycles over shift spaces?

Key findings

• All primitive products of the fully subtractive matrices $ A_{FS,d}^{(k)} $ are Pisot, with the primitivity condition equivalent to all indices $ 1, \dots, d $ appearing in the product sequence.
• For the 3×3 Brun matrices, a product is primitive if and only if $ A_{Br}^{(3)} $ appears in the product, and such products are also Pisot.
• The second largest eigenvalue in absolute value is bounded above by the operator norm on the orthogonal complement of the Perron-Frobenius eigenvector, enabling spectral control.
• The existence of a word $ w $ such that $ \|A^{(w)}\|_{D^{(w)}} < 1 $ implies $ \gamma_2 < 0 $, confirming the Pisot spectrum in the Lyapunov sense.
• The method provides a new, elementary proof for the Pisot property in dimension $ d=3 $, differing from the earlier induction-based proof in [ArIt01].
• The results extend to the Lyapunov spectrum, showing $ \gamma_1 > 0 > \gamma_2 $ under mild ergodicity and log-integrability assumptions, linking to exponential convergence in multidimensional continued fractions.