Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Remarks on the dynamics of the horocycle flow for homogeneous foliations
  by hyperbolic surfaces   This article is a first step towards the understanding of the dynamics of the
horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated
by a question formulated by M. Martinez and A. Verjovsky on the minimality of
this flow assuming that the natural affine foliation is minimal too. We have
tried to offer a simple presentation, which allows us to update and shed light
on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of
the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this
result to the product of PSL(2,R) and a Lie group G, which places us within the
homogeneous framework investigated by M. Ratner. Since our purpose is to deal
with non-homogeneous situations, we do not use Ratner's famous Orbit-Closure
Theorem, but we give an elementary proof. We show that this special situation
arises for homogeneous Riemannian and Lie foliations, reintroducing the
foliation point of view. Examples and counter-examples take an important place
in our work, in particular, the very instructive case of the hyperbolic torus
bundles over the circle. Our aim in writing this text is to offer to the reader
an accessible introduction to a subject that was intensively studied in the
algebraic setting, although there still are unsolved geometric problems.

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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg