On eigen-structures for pseudoAnosov maps

TL;DR

This paper explores the spectral structures of pseudo-Anosov maps, revealing how eigenvalues relate to transverse eigen-cocycles and distributions, and employs symbolic dynamics and the Franks-Shub Theorem for analysis.

Contribution

It introduces a detailed analysis of the hyperbolic spectra of pseudo-Anosov maps, connecting eigenvalues to transverse structures and distributions using symbolic dynamics and existing theorems.

Findings

Eigenvalues correspond to transverse eigen-cocycles and distributions.

Markov transition eigenvalues produce holonomy invariant functions.

The approach links spectral data to symbolic and foliation structures.

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.