Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic behaviour of ergodic integrals of `renormalizable' parabolic
  flows   Ten years ago A. Zorich discovered, by computer experiments on interval
exchange transformations, some striking new power laws for the ergodic
integrals of generic non-exact Hamiltonian flows on higher genus surfaces. In
Zorich's later work and in a joint paper authored by M. Kontsevich, Zorich and
Kontsevich were able to explain conjecturally most of Zorich's discoveries by
relating them to the ergodic theory of Teichm\"uller flows on moduli spaces of
Abelian differentials. In this article, we outline a generalization of the
Kontsevich-Zorich framework to a class of `renormalizable' flows on
`pseudo-homogeneous' spaces. We construct for such flows a `renormalization
dynamics' on an appropriate `moduli space', which generalizes the Teichm\"uller
flow. If a flow is renormalizable and the space of smooth functions is
`stable', in the sense that the Lie derivative operator on smooth functions has
closed range, the behaviour of ergodic integrals can be analyzed, at least in
principle, in terms of an Oseledec's decomposition for a `renormalization
cocycle' over the bundle of `basic currents' for the orbit foliation of the
flow. This approach was suggested by the author's proof of the
Kontsevich-Zorich conjectures and it has since been applied, in collaboration
with L. Flaminio, to prove that the Zorich phenomenon generalizes to several
classical examples of volume preserving, uniquely ergodic, parabolic flows such
as horocycle flows and nilpotent flows on homogeneous 3-manifolds.

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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph