Achievable spectral radii of symplectic Perron-Frobenius matrices

Abstract

A pseudo-Anosov surface automorphism $\phi$ has associated to it an algebraic unit $\lambda_\phi$ called the dilatation of $\phi$. It is known that in many cases $\lambda_\phi$ 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 $\lambda$ appears as an eigenvalue for some integral symplectic matrix. We then show that if $\lambda$ is real and the greatest in modulus of its algebraic conjugates and their inverses, then $\lambda^n$ is the spectral radius of an integral Perron-Frobenius matrix preserving a prescribed symplectic form $L$. An immediate application of this is that for $\lambda$ as above, $\log(\lambda^n)$ is the topological entropy of a subshift of finite type.