  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.
