Atlas: Laminations of the unit disk with irrational rotation gaps by John C. Mayer

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Spring Topology and Dynamical Systems Conference 2008
March 13-15, 2008
University of Wisconsin Milwaukee and Marquette University
Milwaukee, WI, USA

Organizers
Ric Ancel, Karen Brucks, Craig Guilbault, Chris Hruska, Suzanne Hruska, Boris Okun (UWM); Paul Bankston (Marquette); Lois Kailhofer (Alverno College).

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Laminations of the unit disk with irrational rotation gaps
by
John C. Mayer
University of Alabama at Birmingham
Coauthors: William O. Bond and Ross M. Ptacek

In a program begun in the early '80's William Thurston proposed to study the Julia sets of complex polynomials through a combinatorial approach using laminations of the unit disk. Thurston's work was never formally published, but his notes circulated widely. Laminations are topological/combinatorial, rather than analytic, tools. In a Circle Dynamics Seminar extending back to the late 90's, a group at UAB, faculty and a changing cast of students, graduate and undergraduate, have been working on understanding Thurston's work on quadratic laminations and extending it to laminations of higher degree. In this talk we will report preliminarily on part of this work carried out with two first-year graduate students, William Bond and Ross Ptacek.

A lamination L of the unit disk D² is a closed collection of chords of D² that intersect, if at all, in an endpoint of each on the boundary circle S¹. The chords in L are called leaves. A gap G of L is the closure of a component of D²\∪L. The boundary Bd(G) of G is composed of leaves and points of S¹. We parameterize S¹ by [0, 1) in the natural way. Consider the d-tupling map s_d:S¹→S¹ defined by s_d(t)=dt mod 1. The map s_d can be extended to leaves linearly and continuously on ∪L. A lamination is d-invariant if, under s_d, it is fully invariant (forward and backward) on the leaves of L and is gap invariant (which has a long technical definition). A leaf of L is a critical leaf if the images of its endpoints are the same point.

One type of gap that occurs in an invariant lamination is an irrational rotation gap G, which has the properties

Bd(G) contains at least one critical leaf,
Bd(G)∩S¹ is a Cantor set, and
under s_d, Bd(G) maps to itself in circular order.

Thus Bd(G) rotates as critical leaves in Bd(G) collapse to points and other leaves map to leaves. With reference to the corresponding Julia set, such an irrational rotation gap corresponds to a fixed Siegel disk or a fixed Cremer point. The nature of this correspondence is an area of active investigation on several fronts, particularly in case the Julia set is not locally connected. However, in this talk, we confine ourselves to a discussion of some of the peculiarities of laminations containing an irrational rotation gap, and what it may imply for the corresponding Julia sets.

Date received: February 27, 2008