Degree-d-invariant Laminations

Thurston, William P.; Baik, Hyungryul; Gao, Yan; Hubbard, John H.; Lindsey, Kathryn; Tan, Lan; Thurston, Dylan P.

doi:10.2307/j.ctvthhdvv.15

Cited by 8 publications

(11 citation statements)

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“…it is a K(B 4 , 1) (an analogous statement holds in every degree). There are several ways to see this; one elegant method is due to Thurston, and explained in [28]. We shall see a quite different and completely transparent demonstration of this fact in § 9.…”

Section: Degreementioning

confidence: 98%

“…Elaminations are an essential combinatorial tool that will be used throughout the sequel, especially beginning with § 4, so throughout this section we just spell out the basic theory, deferring the connection to dynamics until the sequel. There are some points of contact between elaminations -and in particular the 'collision topology' on the space EL -to the theory partially developed by Thurston in [28]; but there are many points of difference, and it seems pointless to try to force the two theories into a common framework.…”

Section: Elaminationsmentioning

confidence: 99%

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Sausages and Butcher Paper

Calegari¹

2021

Preprint

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For each d > 1 the shift locus of degree d, denoted S d , is the space of normalized degree d polynomials in one complex variable for which every critical point is in the attracting basin of infinity under iteration. It is a complex analytic manifold of complex dimension d − 1.We are able to give an explicit description of S d as a complex of spaces over a contractible Ãd−2 building, and to describe the pieces in two quite different ways:(1) (combinatorial): in terms of dynamical extended laminations; or (2) (algebraic): in terms of certain explicit 'discriminant-like' affine algebraic varieties. From this structure one may deduce numerous facts, including that S d has the homotopy type of a CW complex of real dimension d − 1; and that S 3 and S 4 are K(π, 1)s.The method of proof is rather interesting in its own right. In fact, along the way we discover a new class of complex surfaces (they are complements of certain singular curves in C 2 ) which are homotopic to locally CAT(0) complexes; in particular they are K(π, 1)s.

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Section: Degreementioning

confidence: 98%

Section: Elaminationsmentioning

confidence: 99%

Sausages and Butcher Paper

Calegari¹

2021

Preprint

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“…On S 1 we have a natural equivalence relation ∼ p given by the Thurston lamination of the polynomial p as follows: t 1 ∼ p t 2 whenever γ(t 1 ) = γ(t 2 ). Then the set S 1 / ∼p is homeomorphic to J p and the polynomial p acting on its Julia set J p is topologically conjugate to the angle doubling map on S 1 / ∼p as in [Th] and [Th1].…”

Section: Distance Between Vertical-like Curvesmentioning

confidence: 99%

A structure theorem for semi-parabolic Hénon maps

Radu

Tanase

2019

Advances in Mathematics

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Consider the parameter space P λ ⊂ C 2 of complex Hénon maps H c,a (x, y) = (x 2 + c + ay, ax), a = 0 which have a semi-parabolic fixed point with one eigenvalue λ = e 2πip/q . We give a characterization of those Hénon maps from the curve P λ that are small perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier λ. We prove that there is an open disk of parameters in P λ for which the semi-parabolic Hénon map has connected Julia set J and is structurally stable on J and J + . The Julia set J + has a nice local description: inside a bidisk D r × D r it is a trivial fiber bundle over J p , the Julia set of the polynomial p, with fibers biholomorphic to D r . The Julia set J is homeomorphic to a quotiented solenoid.Theorem 1.1 (Structure Theorem). Let p(x) = x 2 + c 0 be a polynomial with a parabolic fixed point of multiplier λ = e 2πip/q . There exists δ > 0 such that for all parameters (c, a) ∈ P λ with 0 < |a| < δ there exists a homeomorphismThe map ψ depends on a, but we will show in Lemmas 12.7 and 12.8 that all maps ψ are conjugate to each other, for sufficiently small 0 < |a| < δ. Thus it does not matter which one we use and we can assume that the model map is ψ(ζ, z) = p(ζ), ζ − 2 z p (ζ) , for some > 0 independent of a. The function ψ is a solenoidal map in the sense of [HOV1]; it behaves like angle-doubling in the first coordinate, and contracts strongly in the second coordinate.Theorem 1.1 shows that J + ∩ V is a trivial fiber bundle over J p , the Julia set of the parabolic polynomial p(x) = x 2 + c 0 , with fibers biholomorphic to D r . The set J + is laminated by Riemann surfaces isomorphic to C. In fact, the current µ + supported on J + defined by Bedford and Smillie in [BS1] is laminar.

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“…Several people have set out to write supplementary material to the manuscript, based on what they learned from him throughout his seminar and email exchanges with him. The outcome is the manuscript [21].…”

mentioning

confidence: 99%

“…The present article is part of the author's Ph.D thesis. The initial purpose, as suggested by my supervisor Tan Lei, was to fill in the empty section "Hausdorff dimension and growth rate" of W. Thurston's manuscript, as part of the supplementary material in [21]. However, as the research developed, the scope of the work exceeded largely what we had expected.…”

mentioning

confidence: 99%

On Thurston’s core entropy algorithm

Gao

2019

Trans. Amer. Math. Soc.

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The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant extending topological entropy for real maps to complex polynomials, whence providing a new tool to study the parameter space of polynomials. The base is a combinatorial algorithm allowing for the computation of the core entropy given by Thurston but without supplying a proof. In this paper, we will describe his algorithm and prove its validity.

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Degree-d-invariant Laminations

Cited by 8 publications

References 0 publications

Sausages and Butcher Paper

Sausages and Butcher Paper

A structure theorem for semi-parabolic Hénon maps

On Thurston’s core entropy algorithm

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