RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mosc. Math. J.: Year: Volume: Issue: Page: Find

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

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
References: PDF file   HTML file

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
\Bibitem{Glu10} \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} 

• http://mi.mathnet.ru/eng/mmj392
• http://mi.mathnet.ru/eng/mmj/v10/i3/p547

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 109 References: 30