Achievable spectral radii of symplectic Perron-Frobenius matrices

2011/04/13 by Robert Ackermann, Ackermann, Robert · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1104.2642

16 pages

arxiv created 2011/04/13 · openalex publication_date 2011/04/13 · arxiv updated 2011/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A pseudo-Anosov surface automorphism ϕ has associated to it an algebraic unit λ_ϕ called the dilatation of ϕ. It is known that in many cases λ_ϕ appears as the spectral radius of a Perron-Frobenius matrix preserving a symplectic form L. We investigate what algebraic units could potentially appear as dilatations by first showing that every algebraic unit λ appears as an eigenvalue for some integral symplectic matrix. We then show that if λ is real and the greatest in modulus of its algebraic conjugates and their inverses, then λⁿ is the spectral radius of an integral Perron-Frobenius matrix preserving a prescribed symplectic form L. An immediate application of this is that for λ as above, log(λⁿ) is the topological entropy of a subshift of finite type.

Citations

• Representations of monomial algebras have poly-exponential complexities
• Small dilatation pseudo-Anosovs and 3-manifolds

Cited by

• Counting totally real units and eigenvalue patterns in \rmSL_n(\mathbb Z) and \rmSp_2n(\mathbb Z) in thin tubes

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