Area and Hausdorff Dimension of Julia Sets of Entire Functions

Curt McMullen

Type: Article

Publication Date: 1987-03-01

Citations: 219

DOI: https://doi.org/10.2307/2000602

Abstract

We show the Julia set of A sin(z) has positive area and the action of A sin(z) on its Julia set is not ergodic; the Julia set of Aexp(z) has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.

Locations

Digital Access to Scholarship at Harvard (DASH) (Harvard University) - View - PDF
Transactions of the American Mathematical Society - View - PDF

Similar Works

Action	Title	Year	Authors
+ PDF Chat	Area and Hausdorff dimension of Julia sets of entire functions	1987	Curt McMullen
+	HAUSDORFF DIMENSION AND HAUSDORFF MEASURES OF JULIA SETS OF ELLIPTIC FUNCTIONS	2003	Janina Kotus Mariusz Urbański
+	Repelling Periodic Points in the Julia Set	1997	Wilhelm Schwick
+ PDF Chat	Hausdorff dimension and measures on Julia sets of some meromorphic maps	1995	Krzysztof Barański
+	Julia set and ergodicity of meromorphic functions	2002	Wael Bahsoun
+	Area and Hausdorff dimension of the set of accessible points of the Julia sets of $λe^z$ and $λ \sin(z)$	1999	Bogusława Karpińska
+	Hausdorff convergence of Julia sets	1999	Bernd Krauskopf Hartje Kriete
+ PDF Chat	Lebesgue measure of Julia sets and escaping sets of certain entire functions	2018	Walter Bergweiler
+ PDF Chat	On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point	1990	Mariusz Urbański
+	Haar measures and Hausdorff dimensions of 𝑝-adic Julia sets	2016	Joanna Furno
+	A Julia Set That Is Everything	2003	Julia Barnes Lorelei Koss
+	The Julia Set of the Mapping z→zexp(z+μ)	1994	乔建永
+	Lebesgue Measure of Escaping Sets of Entire Functions in the Eremenko-Lyubich Class	2017	Weiwei Cui
+	NON-TRIVIAL DEFORMATION OF AN ENTIRE FUNCTION abz +ebz + c	2001	Shunsuke Morosawa Masahiko Taniguchi
+ PDF Chat	NON-TRIVIAL DEFORMATION OF AN ENTIRE FUNCTION abz +ebz + c	2001	Shunsuke Morosawa Masahiko Taniguchi
+	Zeta functions and topological entropy of periodic nonautonomous dynamical systems	2012	João Ferreira Alves Michal Málek
+	The Hausdorff Dimension of Julia Sets of Entire Functions IV	2000	Gwyneth M. Stallard
+	Some properties of Julia sets of semi-hyperbolic entire functions(Complex Dynamics and its Related Topics)	2007	Masashi Kisaka
+ PDF Chat	Meromorphic functions whose action on their Julia sets is Non-Ergodic	2024	Tao Chen Yunping Jiang Linda Keen
+	Julia sets, Hausdorff dimension and phase transition	2011	Junyang Gao

Works That Cite This (214)

Action	Title	Year	Authors
+ PDF Chat	Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder	2016	Pavel Bleher Mikhail Lyubich Roland K. W. Roeder
+	The failure of Ruelle's property for entire functions	2019	Volker Mayer Anna Zdunik
+ PDF Chat	Schmidt games and non-dense forward orbits of certain partially hyperbolic systems	2015	Weisheng Wu
+	Wiman-Valiron discs and the dimension of Julia sets	2019	James Waterman
+ PDF Chat	On the dimension of points which escape to infinity at given rate under exponential iteration	2021	Krzysztof Barański Bogusława Karpińska
+	Hausdorff Dimension, Its Properties, and Its Surprises	2007	Dierk Schleicher
+ PDF Chat	Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions	2013	Lasse Rempe
+	Hausdorff dimension of the hairs without endpoints for λ exp z	1999	Bogusława Karpińska
+ PDF Chat	On the Hausdorff dimension of the escaping set of certain meromorphic functions	2012	Walter Bergweiler Janina Kotus
+	ERGODICITY OF SOME CLASSES OF MEROMORPHIC FUNCTIONS	1999	Linda Keen Janina Kotus

Works Cited by This (13)

Action	Title	Year	Authors
+	Conformal Invariants: Topics in Geometric Function Theory	1973	Lars V. Ahlfors
+	Conformal dynamical systems	1983	Dennis Sullivan
+ PDF Chat	On iterates of <i>e</i><sup><i>z</i></sup>	1981	Michał Misiurewicz
+ PDF Chat	Some entire functions with multiply-connected wandering domains	1985	I. N. Baker
+	Injectivity of local quasi-isometries	1982	F. W. Gehring
+	On quasi‐isometric mappings, I	1968	Fritz John
+	The domains of normality of an entire function	1975	I. N. Baker
+ PDF Chat	On the measurable dynamics of <i>z</i> → <i>e<sup>z</sup></i>	1985	Étienne Ghys Lisa R. Goldberg Dennis Sullivan
+ PDF Chat	A finiteness theorem for a dynamical class of entire functions	1986	Lisa R. Goldberg Linda Keen
+ PDF Chat	Dynamics of exp (<i>z</i>)	1984	Robert L. Devaney Michał Krych