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

Tiozzo, Giulio

doi:10.1016/j.aim.2014.12.033

Cited by 28 publications

(30 citation statements)

References 31 publications

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“…Earlier we had shown [MeS] that the biaccessibility dimension is always less than 1, except when the Julia set is an interval, strengthening earlier work by Smirnov [Sm], Zakeri [Za], and Zdunik [Z]. Quite recently, Tiozzo wrote an interesting thesis on core entropy and related questions [Ti1], in particular with a relation to the thermodynamic formalism. Some additional history, as well as the relation between core entropy and biaccessibility dimension in the general case, are given in the appendix by Wolf Jung.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 52%

“…Bill Thurston inspired a number of people to investigate core entropy. In particular, there are a survey on current work and open problems by Tan Lei [TL], a manuscript by Wolf Jung [Ju], and two manuscripts by Giulio Tiozzo [Ti1,Ti2].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…along which entropy is continuous, then h is continuous at h: this is the content of Theorem 5.8. By work of Tiozzo [Ti1] and Jung [Ju,Theorem 4.9], this is true for all dyadic angles ϑ.…”

Section: Continuity Of Entropymentioning

confidence: 94%

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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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Section: Introduction and Statement Of Resultsmentioning

confidence: 52%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Core Entropy of Quadratic Polynomials

Dudko

Schleicher

2020

Arnold Math J.

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“…For quadratic polynomials, G.Tiozzo showed the continuity of core-entropy along principal veins of the Mandelbrot set M in [Ti13]. This result is generalized by W. Jung to all veins [Ju13].…”

Section: Motivationmentioning

confidence: 99%

Criterion for rays landing together

Zeng

2020

Trans. Amer. Math. Soc.

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Let f be a polynomial with degree ≥ 2 and the Julia set J f locally connected. We give a partition of complex plane C and show that, if z, z in J f have the same itinerary respect to the partition, then either z = z or both of them lie in the boundary of a Fatou component U , which is eventually iterated to a siegel disk. As an application, we prove the monotonicity of core entropy for the quadratic polynomial family {fc = z 2 + c : fc has no Siegel disks and J fc is locally connected }.

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“…Ashish et al researched features of Mandelbrot set in Noor orbit [3]. Tiozzo computed topological entropy and dimensions of sections of Mandelbrot set [28]. Andreadis and Karakasidis researched approximation of boundary structure in Mandelbrot set [2].…”

Section: Introductionmentioning

confidence: 99%

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

Liu¹,

Zheng²,

Fu³

et al. 2017

J. Nonlinear Sci. Appl.

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Generalized Mandelbrot set (k-M set) is the basis of fractal analysis. This paper presents a novel method to generate k-M set, which generates k-M set precisely by constructing its asymptote family. Correctness of the proposed method is proved as well as computational complexity is researched. Further, application of the generation method is studied, which is used to analyze distribution of boundary points and periodic points of k-M set. Finally, experiments have been implemented to evaluate the theoretical results.

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

Cited by 28 publications

References 31 publications

Core Entropy of Quadratic Polynomials

Core Entropy of Quadratic Polynomials

Criterion for rays landing together

Fractal generation method based on asymptote family of generalized Mandelbrot set and its application

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