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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Laminations in the language of leaves


Authors: Alexander M. Blokh, Debra Mimbs, Lex G. Oversteegen and Kirsten I. S. Valkenburg
Journal: Trans. Amer. Math. Soc. 365 (2013), 5367-5391
MSC (2010): Primary 37F20; Secondary 37F10
Published electronically: April 9, 2013
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Abstract: Thurston defined invariant laminations, i.e. collections of chords of the unit circle $ \mathbb{S}$ (called leaves) that are pairwise disjoint inside the open unit disk and satisfy a few dynamical properties. To be directly associated to a polynomial, a lamination has to be generated by an equivalence relation with specific properties on $ \mathbb{S}$; then it is called a q-lamination. Since not all laminations are q-laminations, then from the point of view of studying polynomials the most interesting are those which are limits of q-laminations. In this paper we introduce an alternative definition of an invariant lamination, which involves only conditions on the leaves (and avoids gap invariance). The new class of laminations is slightly smaller than that defined by Thurston and is closed. We use this notion to elucidate the connection between invariant laminations and invariant equivalence relations on $ \mathbb{S}$.


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Additional Information

Alexander M. Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: ablokh@math.uab.edu

Debra Mimbs
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Address at time of publication: Department of Mathematics, Lee University, Cleveland, Tennessee 37320-3450
Email: dmimbs@leeuniversity.edu

Lex G. Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: overstee@math.uab.edu

Kirsten I. S. Valkenburg
Affiliation: Faculteit der Exacte Wetenschappen, Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
Email: kirstenvalkenburg@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05809-6
PII: S 0002-9947(2013)05809-6
Keywords: Thurston lamination, complex polynomial, Julia set
Received by editor(s): January 18, 2011
Received by editor(s) in revised form: February 14, 2012
Published electronically: April 9, 2013
Additional Notes: The first and third authors were supported in part by NSF-DMS-0901038 and NSF-DMS-0906316. The fourth author was supported by the Netherlands Organization for Scientific Research (NWO), under grant 613.000.551; she also thanks the Department of Mathematics at the University of Alabama at Birmingham for its hospitality.
Dedicated: This paper is dedicated to the memory of Bill Thurston
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.