Laminations in the language of leaves

Blokh, Alexander; Mimbs, Debra; Oversteegen, Lex G.; Valkenburg, Kirsten I. S.

doi:10.1090/s0002-9947-2013-05809-6

Cited by 23 publications

(68 citation statements)

References 7 publications

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“…Laminations with properties (1)-(3) are called quadratic invariant laminations. By [BMOV13] all quadratic q-laminations L ∼ are invariant, however the converse is not true and there are quadratic invariant laminations that are not q-laminations. Below we often call quadratic invariant laminations simply quadratic laminations.…”

Section: Figure 1 the Geolamination Qmlmentioning

confidence: 99%

“…A lamination L ∼c thus obtained satisfies certain dynamical properties (in our presentation we rely upon [BMOV13]). Below we think of σ 2 applied to a chord with endpoints a and b so that it maps to the chord whose endpoints are σ 2 (a) and σ 2 (b); we can think of this as an extension of σ 2 over and make it linear on .…”

Section: Figure 1 the Geolamination Qmlmentioning

confidence: 99%

“…Indeed, consider all pullbacks of m 0 in L(m) and all limit leaves of such pullbacks. By [BMOV13], this collection L of leaves is a lamination, and by construction L ⊂ L(m). Since L(m) is almost non-renormalizable, L = L. Hence, pullbacks of m 0 in L(m) approximate m. By Lemma 4.3, the minor m is approximated by offsprings of m 0 .…”

Section: Proofmentioning

confidence: 99%

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Dynamical generation of parameter laminations

Blokh¹,

Oversteegen²,

Timorin

2020

Dynamics: Topology and Numbers

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Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of "pullbacks" of minors from the main cardioid.

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Section: Figure 1 the Geolamination Qmlmentioning

confidence: 99%

Section: Figure 1 the Geolamination Qmlmentioning

confidence: 99%

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Dynamical generation of parameter laminations

Blokh¹,

Oversteegen²,

Timorin

2020

Dynamics: Topology and Numbers

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“…Moreover, these siblings can be chosen to be disjoint from the leaf. Definition 2.2 implies Thurston's but is slightly more restrictive [BMOV13].…”

Section: Laminations and Their Propertiesmentioning

confidence: 99%

“…Theorem 2.5 (Theorem 3.21 [BMOV13]). The family of all invariant geodesic laminations L is compact in C(C(D)).…”

Section: Laminations and Their Propertiesmentioning

confidence: 99%

Models for spaces of dendritic polynomials

Blokh

Oversteegen

Ptacek

et al. 2019

Trans. Amer. Math. Soc.

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Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.

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Perfect subspaces of quadratic laminations

2018

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The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to.

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Laminations in the language of leaves

Cited by 23 publications

References 7 publications

Dynamical generation of parameter laminations

Dynamical generation of parameter laminations

Models for spaces of dendritic polynomials

Perfect subspaces of quadratic laminations

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