Dilatations and continued fractions

Abstract

We show how to express the eigenvalue of maximum modulus and the corresponding eigenvector of a certain class of Perron-Frobenius matrices in continued fraction form, using generalized continued fractions in the sense of Jacobi and Perron. As an application we show that if ψ is a pseudo-Anosov homeomorphism of a hyperbolic surface such that for an appropriate track τ (a slight generalization of the track), ψ(τ) collapses to τ with Perron-Frobenius incidence matrix, then after possibly iterating ψ, say L times, the invariant measured foliation (represented in vector form using the theory of tracks) and the dilatation of ψL can be expressed in generalized continued fraction form. As another application, we show how to describe the invariant axes of the hyperbolic elements (after normalization) in PSL(2, Z) using (ordinary) continued fractions.