scholarly journals Repellers for real analytic maps

1982 ◽  
Vol 2 (1) ◽  
pp. 99-107 ◽  
Author(s):  
David Ruelle
Keyword(s):  
Hausdorff Dimension ◽  
Rational Function ◽  
General Result ◽  
Julia Set ◽  
Real Analytic ◽  
Analytic Maps

AbstractThe purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see corollary 5).

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 $\dim J(f) = d$.

Hausdorff dimension of Julia sets of complex Hénon mappings

1996 ◽  
Vol 16 (4) ◽  
pp. 849-861 ◽  
Author(s):  
A. Verjovsky ◽  
H. Wu
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Invariant Sets ◽  
Analytic Dependence ◽  
Fractal Sets ◽  
Complex Dimension ◽  
Real Analytic

AbstractThe Hausdorff dimension of closed invariant sets under diffeomorphisms is an interesting concept as it is a measure of their complexity. The theory of holomorphic dynamical systems provides us with many examples of fractal sets and, in particular, a theorem of Ruelle [Ru1] shows that the Hausdorff dimension of the Julia set depends real analytically onfiffis a rational function of ℂ and the Julia setJoffis hyperbolic. In this paper we generalize Ruelle's result for complex dimension two and show the real analytic dependence of the Hausdorff dimension of the corresponding Julia sets of hyperbolic Hénon mappings.

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).

scholarly journals Real analyticity of Hausdorff dimension for expanding rational semigroups

2009 ◽  
Vol 30 (2) ◽  
pp. 601-633 ◽  
Author(s):  
HIROKI SUMI ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Hausdorff Dimension ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Maps ◽  
Dimension Function ◽  
Analytic Family ◽  
Open Set ◽  
Extensive Collection ◽  
Real Analytic ◽  
Parameter Function

AbstractWe consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the Hausdorff dimension of the Julia set is a real-analytic and plurisubharmonic function of the parameter. Moreover, we provide an extensive collection of examples of analytic families of semigroups satisfying all of the above conditions and we analyze in detail the corresponding Bowen’s parameters and Hausdorff dimension function.

The Hausdorff Dimension is Convex on the Left Side of 1/4

2017 ◽  
Vol 60 (4) ◽  
pp. 911-936 ◽  
Author(s):  
Ludwik Jaksztas
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Mandelbrot Set ◽  
Real Analytic

AbstractLet d(c) denote the Hausdorff dimension of the Julia set Jc of the polynomial fc(z) = z2 +c. The function c ↦ d(c) is real-analytic on the interval (–3/4, 1/4), which is in the domain bounded by the main cardioid of the Mandelbrot set. We prove that the function d is convex close to 1/4 on the left side of it.

Geometry and ergodic theory of conformal non-recurrent dynamics

1997 ◽  
Vol 17 (6) ◽  
pp. 1449-1476 ◽  
Author(s):  
MARIUSZ URBAŃSKI
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Rational Function ◽  
Ergodic Theory ◽  
Critical Points ◽  
Julia Set ◽  
Sufficient Conditions ◽  
Infinite Measure ◽  
Conformal Measure ◽  
Box Dimensions

Let $h$ be the Hausdorff dimension of the Julia set of a rational function $T$ with no non-periodic recurrent critical points and let $m$ be the only $h$-conformal measure for $T$. We prove the existence of a $\sigma$-finite $T$-invariant measure $\mu$ equivalent with $m$. The measure $\mu$ is then proved to be ergodic and conservative and we study the set of those points whose all open neighborhoods have infinite measure $\mu$. Developing the concept of the inverse jump transformation we show that the packing and Hausdorff dimensions of the conformal measure are equal to $h$. We also provide some sufficient conditions for Hausdorff and box dimensions of the Julia set to be equal.

scholarly journals On the Lê-Milnor fibration for real analytic maps

2016 ◽  
Vol 290 (2-3) ◽  
pp. 382-392 ◽  
Author(s):  
Aurélio Menegon Neto ◽  
José Seade
Keyword(s):  
Milnor Fibration ◽  
Real Analytic ◽  
Analytic Maps

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

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.

scholarly journals Any counterexample to Makienko’s conjecture is an indecomposable continuum

2009 ◽  
Vol 29 (3) ◽  
pp. 875-883 ◽  
Author(s):  
CLINTON P. CURRY ◽  
JOHN C. MAYER ◽  
JONATHAN MEDDAUGH ◽  
JAMES T. ROGERS Jr
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Fatou Set ◽  
Indecomposable Continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

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