Hausdorff dimension 2 for Julia sets of quadratic polynomials

2001 ◽  
Vol 237 (3) ◽  
pp. 571-583 ◽  
Author(s):  
Stefan-M. Heinemann ◽  
Bernd O. Stratmann
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Quadratic Polynomials ◽  
Dimension 2

Conformal trace theorem for Julia sets of quadratic polynomials

2017 ◽  
Vol 39 (9) ◽  
pp. 2481-2506 ◽  
Author(s):  
A. CONNES ◽  
E. MCDONALD ◽  
F. SUKOCHEV ◽  
D. ZANIN
Keyword(s):  
Hausdorff Dimension ◽  
Jordan Curve ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Trace Theorem ◽  
Quadratic Polynomials ◽  
Full Proof

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

HAUSDORFF DIMENSION OF JULIA SETS OF QUADRATIC POLYNOMIALS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850020
Author(s):  
LUIS MANUEL MARTÍNEZ ◽  
GAMALIEL BLÉ
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Small Oscillation ◽  
Expanding Maps ◽  
Quadratic Polynomials

The Hausdorff dimension of Julia sets of expanding maps can be computed by the eigenvalue algorithm. In this work, an implementation of this algorithm for quadratic polynomial, that allows the calculation of the Hausdorff dimension of Julia sets for complex parameters, is done. In particular, the parameters in a neighborhood of the parabolic parameter [Formula: see text] are analyzed and a small oscillation in Hausdorff dimension is shown.

Hausdorff Dimension of Julia Sets

2019 ◽  
pp. 153-192
Author(s):  
Xin-Hou Hua ◽  
Chung-Chun Yang
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

scholarly journals Shadowing and -limit sets of circular Julia sets

2013 ◽  
Vol 35 (4) ◽  
pp. 1045-1055 ◽  
Author(s):  
ANDREW D. BARWELL ◽  
JONATHAN MEDDAUGH ◽  
BRIAN E. RAINES
Keyword(s):  
Unit Circle ◽  
Complex Plane ◽  
Julia Sets ◽  
Closed Set ◽  
Quadratic Polynomials ◽  
Limit Sets ◽  
Kneading Sequence

AbstractIn this paper we consider quadratic polynomials on the complex plane${f}_{c} (z)= {z}^{2} + c$and their associated Julia sets,${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an$n$-tupling. In this case${J}_{c} $contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that${f}_{c} : {J}_{c} \rightarrow {J}_{c} $has shadowing, and we classify all$\omega $-limit sets for these maps by showing that a closed set$R\subseteq {J}_{c} $is internally chain transitive if, and only if, there is some$z\in {J}_{c} $with$\omega (z)= R$.

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