Density of hyperbolicity for rational maps with Cantor Julia sets

2011 ◽  
Vol 32 (5) ◽  
pp. 1711-1726 ◽  
Author(s):  
WENJUAN PENG ◽  
YONGCHENG YIN ◽  
YU ZHAI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

scholarly journals Rational maps whose Julia sets are Cantor circles

2013 ◽  
Vol 35 (2) ◽  
pp. 499-529 ◽  
Author(s):  
WEIYUAN QIU ◽  
FEI YANG ◽  
YONGCHENG YIN
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Cantor Set ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

scholarly journals On (non-)local connectivity of some Julia sets

2014 ◽  
Author(s):  
Alexandre Dezotti ◽  
Pascale Roesch
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complete Characterization ◽  
Rational Maps ◽  
Local Connectivity ◽  
Rational Map ◽  
Critical Set ◽  
Quadratic Polynomials ◽  
Non Local

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.

scholarly journals THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS

2002 ◽  
Vol 85 (2) ◽  
pp. 467-492 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Julia Set ◽  
Density Theorem ◽  
Finite Union ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Open Set ◽  
Almost All ◽  
Scenery Flow ◽  
Measure Of Maximal Entropy

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

CONTROL AND SYNCHRONIZATION OF JULIA SETS OF THE COMPLEX PERTURBED RATIONAL MAPS

2013 ◽  
Vol 23 (05) ◽  
pp. 1350083 ◽  
Author(s):  
YONGPING ZHANG
Keyword(s):  
Optimal Control ◽  
Function Method ◽  
Control Method ◽  
Control Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Control And Synchronization ◽  
Function Synchronization ◽  
Optimal Control Function

The dynamical and fractal behaviors of the complex perturbed rational maps [Formula: see text] are discussed in this paper. And the optimal control function method is taken on the Julia set of this system. In this control method, infinity is regarded as a fixed point to be controlled. By substituting the driving item for an item in the optimal control function, synchronization of Julia sets of two such different systems is also studied.

Transitive maps which are not ergodic with respect to Lebesgue measure

1996 ◽  
Vol 16 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Sebastian Van Strien
Keyword(s):  
Lebesgue Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Iteration Scheme ◽  
Rational Maps ◽  
Positive Lebesgue Measure ◽  
Rational Map ◽  
Interval Maps ◽  
Open Set ◽  
Totally Disconnected

AbstractIn this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

An exceptional set in the ergodic theory of rational maps of the Riemann sphere

1997 ◽  
Vol 17 (2) ◽  
pp. 253-267 ◽  
Author(s):  
A. G. ABERCROMBIE ◽  
R. NAIR
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Hausdorff Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Maps ◽  
Exceptional Set ◽  
Rational Map ◽  
Pointwise Ergodic Theorem ◽  
Conformal Measure

A rational map $T$ of degree not less than two is known to preserve a measure, called the conformal measure, equivalent to the Hausdorff measure of the same dimension as its Julia set $J$ and supported there, with respect to which it is ergodic and even exact. As a consequence of Birkhoff's pointwise ergodic theorem almost every $z$ in $J$ with respect to the conformal measure has an orbit that is asymptotically distributed on $J$ with respect to this measure. As a counterpoint to this, the following result is established in this paper. Let $\Omega(z)=\Omega_{T}(z)$ denote the closure of the set $\{T^{n}(z):n=1,2,\ldots\}$. For any expanding rational map $T$ of degree at least two we set \[ S(z_{0})=\{z\in J:z_{0}\not\in \Omega_{T}(z)\}. \] We show that for all $z_{0}$ the Hausdorff dimensions of $S(z)$ and $J$ are equal.

scholarly journals Jarník and Julia; a Diophantine analysis for parabolic rational maps for Geometrically Finite Kleinian Groups with Parabolic Elements

2002 ◽  
Vol 91 (1) ◽  
pp. 27 ◽  
Author(s):  
B. O. Stratmann ◽  
M. Urbański
Keyword(s):  
Number Theory ◽  
Multifractal Analysis ◽  
Julia Sets ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Rational Map ◽  
Geometrically Finite ◽  
Conformal Measure ◽  
Further Development

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarník in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a "weak multifractal analysis" of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan's famous dictionary translating between the theory of Kleinian groups and the theory of rational maps.

Geometric measures for parabolic rational maps

1992 ◽  
Vol 12 (1) ◽  
pp. 53-66 ◽  
Author(s):  
M. Denker ◽  
M. Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Rational Maps ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Conformal Measure

AbstractLet h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

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.

-stability of expanding maps in non-Archimedean dynamics

2017 ◽  
Vol 39 (4) ◽  
pp. 1002-1019
Author(s):  
JUNGHUN LEE
Keyword(s):  
Characteristic Zero ◽  
Complex Dynamics ◽  
Julia Set ◽  
Rational Maps ◽  
Expanding Maps ◽  
Rational Map

The aim of this paper is to show $J$-stability of expanding rational maps over an algebraically closed, complete and non-Archimedean field of characteristic zero. More precisely, we will show that for any expanding rational map, there exists a neighborhood of it such that the dynamics on the Julia set of any rational map in the neighborhood is the same as the dynamics of the expanding rational map as a non-Archimedean analogue of a corollary of Mañé, Sad and Sullivan’s result [On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4)16 (1983), 193–217] in complex dynamics.

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