Achievable spectral radii of symplectic Perron–Frobenius matrices

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 λ 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 (λ) is the topological entropy of a subshift of finite type.