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Dilatations and continued fractions

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
174
Identifiers
DOI: 10.1016/0024-3795(92)90051-b

Abstract

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 ψ L 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.

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