New York Journal of Mathematics
Volume 17 (2011) 683-697


R. Ackermann

Achievable spectral radii of symplectic Perron-Frobenius matrices

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Published: October 7, 2011
Keywords: Perron-Frobenius matrix, spectral radii, pseudo-Anosov, dilatation, subshift of finite type
Subject: 57M99

A pseudo-Anosov surface automorphism ϕ has associated to it an algebraic unit λϕ called the dilatation of ϕ. It is known that in many cases λϕ 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.

Author information

Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106