Achievable spectral radii of symplectic Perron-Frobenius matrices

作者：Ackermann R^*

来源：The New York Journal of Mathematics, 2011, 17: 683-697.

• Perron-Frobenius matrix
• spectral radii
• pseudo-Anosov
• dilatation
• subshift of finite type

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 asymplectic 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.

• 出版日期2011