Hausdorff dimension of biaccessible angles for quadratic polynomials

Bruin, Henk; Schleicher, Dierk

doi:10.4064/fm276-6-2016

Cited by 9 publications

(9 citation statements)

References 22 publications

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“…More recent work on biaccessible points is due, among others, to Zakeri [47] and Zdunik [48]. The first equality in Theorem 1.1 has also been established independently by Bruin and Schleicher [9].…”

Section: Introductionmentioning

confidence: 72%

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Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Tiozzo

2015

Advances in Mathematics

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A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy. Indeed, we prove an "entropy formula" relating the entropy of a polynomial restricted to its Hubbard tree to the Hausdorff dimension of the set of rays landing on the corresponding vein in the Mandelbrot set. The results contribute to the recent program of W. Thurston of understanding the geometry of the Mandelbrot set via the core entropy.

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Section: Introductionmentioning

confidence: 72%

“…Moreover, since we are mostly interested in principal veins, we will treat in detail only the case of trees with particular topological types. An alternative, independent approach to continuity is in [9].…”

Section: Kneading Theory For Hubbard Treesmentioning

confidence: 99%

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Tiozzo

2015

Advances in Mathematics

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“…The biaccessibility dimension is given as the Hausdorff dimension of those external angles whose the corresponding rays land together with some other ray. In [13], a discussion about a purely combinatorial characterization of biaccessible angles provides for both dynamic and parameter forms, and the Hausdorff dimension of the set of biaccessible angles estimate here. Only the case of quadratic polynomials p c = z 2 + c is considered in this work.…”

Section: Different Approaches To Studying Dynamics Of One Variable Co...mentioning

confidence: 99%

On Some Advances in Dynamics of One Variable Complex Functions

Sajid¹

2022

Int. J. of Appl. Math.

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In this article, we present some recent advances in the dynamics of one variable complex functions which are especially associated to the Julia sets, the Fatou sets, the Sigel disks, the Herman rings, fixed points as well as the Mandelbrot sets. For the sake of interest, we consider mainly research works which have been done from 2015 to 2019. To achieve our purpose, some interesting and selected themes are considered to show some recent advances in the dynamics of one variable complex functions.

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“…For the opposite estimate, note that the plane is cut into pieces successively by precritical ray pairs, and the angles of a piece of level n form up to n intervals of total length 2 −n according to [BrSc2,Lemma 4.1].…”

Section: Suppose That Eithermentioning

confidence: 99%

“…For the record, we observe that along the way we proved that core entropy h is strictly monotone on arcs before dyadic endpoints: if c is dyadic and c ≺ c , then h(c) < h(c ) (the general result in Lemma 2.5 would only give h(c) ≤ h(c )). In fact, there are parameters c ≺ c so that entropy is constant along [c, c ]; this happens when c and c are within the same little Mandelbrot sets, for instance within the "main molecule of M" (which was shown in [BrSc2] to be the locus of parameters with zero biaccessibility dimension).…”

Section: Irrational Angles and The Tiozzo Conjecturementioning

confidence: 99%

Core Entropy of Quadratic Polynomials

Dudko

Schleicher

2020

Arnold Math J.

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We give a combinatorial definition of "core entropy" for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the α fixed point from its negative). This notion extends known definitions that work in cases when the polynomial is postcritically finite or when the topology of the Julia set has good properties, and it applies to all quadratic polynomials in the Mandelbrot set.We prove that core entropy is continuous as a function of the complex parameter. In fact, we model the Julia set as an invariant quadratic lamination in the sense of Thurston: this depends on the external angle of a parameter in the boundary of the Mandelbrot set, and one can define core entropy directly from the angle in combinatorial terms. As such, core entropy is continuous as a function of the external angle.Moreover, we prove a conjecture of Giulio Tiozzo about local and global maxima of core entropy as a function of external angles: local maxima are exactly dyadic angles, and the unique global maximum within any wake occurs at the dyadic angle of lowest denominator. We also describe where local minima occur.An appendix by Wolf Jung relates different concepts of core entropy and biaccessibility dimension and thus shows that biaccessibility dimension is continuous as well.

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Hausdorff dimension of biaccessible angles for quadratic polynomials

Cited by 9 publications

References 22 publications

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

On Some Advances in Dynamics of One Variable Complex Functions

Core Entropy of Quadratic Polynomials

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