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Mosc. Math. J., 2010, Volume 10, Number 3, Pages 547–591 (Mi mmj392)  

This article is cited in 1 scientific paper (total in 1 paper)

On density of horospheres in dynamical laminations

A. Glutsyukab

a Poncelet Laboratory (UMI of CNRS and Independent University of Moscow), Moscow, Russia
b CNRS, Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, Lyon, France

Abstract: Sullivan's dictionary relates two domains of complex dynamics: Kleinian groups and rational iterations on the Riemann sphere. In 1997 M. Lyubich and Y. Minsky have extended the Sullivan's dictionary by constructing an analogue of the hyperbolic manifold of a Kleinian group: the so-called quotient hyperbolic lamination associated to a rational function. This is an abstract topological space constructed from the space of backward orbits of the rational function that carries a “foliation” (more precisely, lamination) by hyperbolic 3-manifolds (that may be singular). The hyperbolic leaves are dense, may be after deleting at most finite number of isolated leaves. Each hyperbolic leaf is foliated by horospheres, which form the unstable foliation (horospheric lamination) for the leafwise vertical geodesic flow. We consider the total laminated space with isolated hyperbolic leaves deleted. We prove that the horospheric lamination is topologically transitive (and there are a lot of dense horospheres), if and only if the corresponding rational function does not belong to the following list of exceptions: powers, Chebyshev polynomials, Lattès examples. We show that the horospheric lamination is minimal, if the corresponding function does not belong to the same list of exceptions and is critically nonrecurrent without parabolics.

Key words and phrases: rational function, natural extension, repelling periodic orbit, affine lamination, hyperbolic lamination, horosphere, minimality.

Full text: http://www.ams.org/.../abst10-3-2010.html
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Bibliographic databases:

Document Type: Article
MSC: 58F23, 57M50
Received: November 9, 2009; in revised form April 21, 2010
Language: English

Citation: A. Glutsyuk, “On density of horospheres in dynamical laminations”, Mosc. Math. J., 10:3 (2010), 547–591

Citation in format AMSBIB
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\by A.~Glutsyuk
\paper On density of horospheres in dynamical laminations
\jour Mosc. Math.~J.
\yr 2010
\vol 10
\issue 3
\pages 547--591
\mathnet{http://mi.mathnet.ru/mmj392}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2732573}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000281878900004}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Glutsyuk A., “Unique ergodicity of horospheric foliations revisited”, J. Fixed Point Theory Appl., 8:1 (2010), 113–149  crossref  mathscinet  zmath  isi  scopus
  • Moscow Mathematical Journal
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