Calculating Hausdorff dimension of Julia sets and Kleinian limit sets

2002 ◽  
Vol 124 (3) ◽  
pp. 495-545 ◽  
Author(s):  
Oliver Jenkinson ◽  
Mark Pollicott
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets ◽  
Limit Sets

Length distortion and the Hausdorff dimension of limit sets

2000 ◽  
Vol 122 (3) ◽  
pp. 465-482 ◽  
Author(s):  
Martin Bridgeman ◽  
Edward C. Taylor
Keyword(s):  
Hausdorff Dimension ◽  
Limit Sets ◽  
Length Distortion

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

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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