Achievable spectral radii of symplectic Perron-Frobenius matrices

TL;DR

This paper explores which algebraic units can serve as spectral radii of symplectic Perron-Frobenius matrices, linking algebraic properties to dynamical systems and topological entropy.

Contribution

It proves that every algebraic unit can be an eigenvalue of some integral symplectic matrix and identifies conditions under which powers of certain units are spectral radii of Perron-Frobenius matrices.

Findings

Every algebraic unit appears as an eigenvalue of an integral symplectic matrix.

If an algebraic unit is real and dominant, its power can be the spectral radius of a Perron-Frobenius matrix.

The logarithm of such spectral radii relates to topological entropy of subshifts.

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.