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On eigen-structures for pseudoAnosov maps | | Philip Boyland
; | | Date: |
15 Sep 2010 | | 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. | | Source: | arXiv, 1009.2932 | | Services: | Forum | Review | PDF | Favorites | |
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