On eigen-structures for pseudoAnosov maps

Abstract

We investigate various structures associated with the hyperbolic Markov and homological spectra of a pseudoAnosov map $\phi$ on a surface. Each unstable eigenvalue of the action of $\phi$ on first cohomolgy yields an eigen-cocycle that is transverse and holonomy invariant to the stable foliation $\mathcal{F}^s$ of $\phi$. Each unstable eigenvalue $\mu$ of a Markov transition matrix for $\phi$ yields a holonomy invariant additive function $G$ on transverse arcs to $\cF^s$ with $\phi^* G = \mu G$. Except when $\mu$ is the dilation of $\phi$, these transverse arc functions do not yield measures, but rather holonomy invariant eigen-distributions which are dual to H\"older functions. Stable homological and Markov eigenvalues yield analogous transverse structures to the unstable foliation of $\phi$. The main tool for working with the homological spectrum is the Franks-Shub Theorem which holds for a general manifold and map. For the Markov spectrum we use the correspondence of the leaf space of stable foliation with a one-sided subshift of finite type. This identification allows the symbolic analog of a transverse arc function to be defined, analyzed, and applied.