scholarly journals Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets

2021 ◽  
pp. 1
Author(s):  
Weiyuan Qiu ◽  
Fei Yang
Keyword(s):  
Julia Sets ◽  
Hausdorff Dimensions

scholarly journals The Hausdorff dimension of Julia sets of entire functions

1991 ◽  
Vol 11 (4) ◽  
pp. 769-777 ◽  
Author(s):  
Gwyneth M. Stallard
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Entire Functions ◽  
Julia Sets ◽  
Hausdorff Dimensions ◽  
Transcendental Entire Functions

AbstractWe construct a set of transcendental entire functions such that the Hausdorff dimensions of the Julia sets of these functions have greatest lower bound equal to one.

Asymptotics of the Hausdorff dimensions of the Julia sets of McMullen maps with error bounds*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 787-816
Author(s):  
Hongbin Lu ◽  
Weiyuan Qiu ◽  
Fei Yang
Keyword(s):  
Asymptotic Behavior ◽  
Hausdorff Dimension ◽  
Error Estimation ◽  
Error Bounds ◽  
Julia Set ◽  
Julia Sets ◽  
Cantor Set ◽  
Hausdorff Dimensions

Abstract For McMullen maps f λ (z) = z p + λ/z p , where λ ∈ C \ { 0 } , it is known that if p ⩾ 3 and λ is small enough, then the Julia set J(f λ ) of f λ is a Cantor set of circles. In this paper we show that the Hausdorff dimension of J(f λ ) has the following asymptotic behavior dim H J ( f λ ) = 1 + log 2 log p + O ( | λ | 2 − 4 / p ) , as λ → 0 . An explicit error estimation of the remainder is also obtained. We also observe a ‘dimension paradox’ for the Julia set of Cantor set of circles.

FIXED POINTS-JULIA SETS

2020 ◽  
Vol 9 (9) ◽  
pp. 6759-6763
Author(s):  
G. Subathra ◽  
G. Jayalalitha
Keyword(s):  
Fixed Points ◽  
Julia Sets

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

scholarly journals Locally connected models for Julia sets

2011 ◽  
Vol 226 (2) ◽  
pp. 1621-1661 ◽  
Author(s):  
Alexander M. Blokh ◽  
Clinton P. Curry ◽  
Lex G. Oversteegen
Keyword(s):  
Julia Sets

On the Hausdorff dimensions of distance sets

Mathematika ◽  
1985 ◽  
Vol 32 (2) ◽  
pp. 206-212 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Hausdorff Dimensions ◽  
Distance Sets
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