کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4582337 1333790 2015 21 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces
ترجمه فارسی عنوان
اظهار نظر در مورد پویایی جریان حلقوی برای صفحات همگن توسط سطوح هذلولی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
چکیده انگلیسی

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. Martínez 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 GG, 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 solvable manifold TA3. 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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Expositiones Mathematicae - Volume 33, Issue 4, 2015, Pages 431–451
نویسندگان
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