 

R. Ackermann
Achievable spectral radii of symplectic PerronFrobenius matrices view print


Published: 
October 7, 2011 
Keywords: 
PerronFrobenius matrix, spectral radii, pseudoAnosov, dilatation, subshift of finite type 
Subject: 
57M99 


Abstract
A pseudoAnosov 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 PerronFrobenius 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 λ^{n} is the spectral radius of an integral PerronFrobenius matrix preserving a prescribed symplectic form L. An immediate application of this is that for λ as above, log (λ^{n}) is the topological entropy of a subshift of finite type.


Author information
Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106
rackermann@math.ucsb.edu

