scholarly journals Hausdorff dimension of biaccessible angles for quadratic polynomials

2017 ◽  
Vol 238 (3) ◽  
pp. 201-239 ◽  
Author(s):  
Henk Bruin ◽  
Dierk Schleicher
Keyword(s):  
Hausdorff Dimension ◽  
Quadratic Polynomials

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 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

scholarly journals The bifurcation locus for numbers of bounded type

2021 ◽  
pp. 1-31
Author(s):  
CARLO CARMINATI ◽  
GIULIO TIOZZO
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Measure Zero ◽  
Period Doubling ◽  
Unit Interval ◽  
Continued Fraction Expansion ◽  
Quadratic Polynomials ◽  
The Family ◽  
Real Quadratic ◽  
Period Doubling Bifurcations

Abstract We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$ , and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$ . The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$ .

scholarly journals Distribution of polynomials with cycles of a given multiplier

2011 ◽  
Vol 201 ◽  
pp. 23-43 ◽  
Author(s):  
Giovanni Bassanelli ◽  
François Berteloot
Keyword(s):  
Quadratic Polynomials ◽  
Bifurcation Locus

AbstractIn the space of degreedpolynomials, the hypersurfaces defined by the existence of a cycle of periodnand multipliereiθare known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

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