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

MathSciNet review:
3074377

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Abstract | References | Similar Articles | Additional Information

Abstract: Thurston defined invariant laminations, i.e. collections of chords of the unit circle (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 ; 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 .

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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

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.