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.

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 $\dim J(f) = d$.

DIMENSIONS OF JULIA SETS OF HYPERBOLIC MEROMORPHIC FUNCTIONS

2001 ◽  
Vol 33 (6) ◽  
pp. 689-694 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Meromorphic Function ◽  
Box Dimensions

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

The Hausdorff dimension of Julia sets of entire functions II

1996 ◽  
Vol 119 (3) ◽  
pp. 513-536 ◽  
Author(s):  
Gwyneth M. Stallard
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Bounded Set

AbstractLetfbe a transcendental entire function such that the finite singularities of f−1lie in a bounded set. We show that the Hausdorff dimension of the Julia set of such a function is strictly greater than one.

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

CONVERGENCE OF JULIA SETS IN THE APPROXIMATION OF λez BY λ[1+(z/d)]d

1993 ◽  
Vol 03 (02) ◽  
pp. 257-270 ◽  
Author(s):  
BERND KRAUSKOPF
Keyword(s):  
Periodic Orbit ◽  
Julia Set ◽  
Julia Sets ◽  
Hausdorff Metric ◽  
Cantor Set ◽  
Compact Sets

The polynomials Pd,λ(z)≔λ[1+(z/d)]dconverge uniformly on compact sets to Eλ(z)≔λez. What this convergence means for the dynamics of these functions when iterated was first studied in Devaney et al. [preprint]. Here we show the convergence of the corresponding Julia sets in the Hausdorff metric for two cases: (1) for λ such that Eλ has an attracting periodic orbit, in which case its Julia set is a Cantor set of curves, and (2) for λ such that the Julia set of Eλ is the whole plane [Formula: see text]. Finally, we give the key ideas of the algorithms designed to illustrate this convergence.

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.

scholarly journals Julia sets of uniformly quasiregular mappings are uniformly perfect

2011 ◽  
Vol 151 (3) ◽  
pp. 541-550 ◽  
Author(s):  
ALASTAIR N. FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Hausdorff Dimension ◽  
Analogous Result ◽  
Julia Set ◽  
Julia Sets ◽  
Quasiregular Mapping ◽  
Higher Dimensions ◽  
Ring Domain ◽  
Rational Map ◽  
Quasiregular Mappings ◽  
Uniformly Perfect

AbstractIt is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

scholarly journals Dynamics of entire functions near the essential singularity

1986 ◽  
Vol 6 (4) ◽  
pp. 489-503 ◽  
Author(s):  
Robert L. Devaney ◽  
Folkert Tangerman
Keyword(s):  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Growth Conditions ◽  
Cantor Set ◽  
Essential Singularity ◽  
Shift Automorphism

AbstractWe show that entire functions which are critically finite and which meet certain growth conditions admit ‘Cantor bouquets’ in their Julia sets. These are invariant subsets of the Julia set which are homeomorphic to the product of a Cantor set and the line [0, ∞). All of the curves in the bouquet tend to ∞ in the same direction, and the map behaves like the shift automorphism on the Cantor set. Hence the dynamics near ∞ for these types of maps may be analyzed completely. Among the entire maps to which our methods apply are exp (z), sin (z), and cos (z).

The Hausdorff dimension of Julia sets of entire functions III

1997 ◽  
Vol 122 (2) ◽  
pp. 223-244 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions

It is known that, for a transcendental entire function f, the Hausdorff dimension of the Julia set of f satisfies 1[les ]dim J(f)[les ]2. In this paper we introduce a family of transcendental entire functions fp, K for which the set {dim J(fp, K)[ratio ]0<p, K<∞} has infemum 1 and supremum 2.

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