scholarly journals Invariant Jordan curves of Sierpiński carpet rational maps

2016 ◽  
Vol 38 (2) ◽  
pp. 583-600 ◽  
Author(s):  
YAN GAO ◽  
PETER HAÏSSINSKY ◽  
DANIEL MEYER ◽  
JINSONG ZENG
Keyword(s):  
Jordan Curve ◽  
Julia Set ◽  
Rational Maps ◽  
Rational Map ◽  
Sierpinski Carpet ◽  
Jordan Curves ◽  
Postcritically Finite ◽  
Sierpiński Carpet

In this paper, we prove that if $R:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer $n_{0}$, such that, for any $n\geq n_{0}$, there exists an $R^{n}$-invariant Jordan curve $\unicode[STIX]{x1D6E4}$ containing the postcritical set of $R$.

Rational maps whose Fatou components are Jordan domains

1996 ◽  
Vol 16 (6) ◽  
pp. 1323-1343 ◽  
Author(s):  
Kevin M. Pilgrim
Keyword(s):  
Jordan Curve ◽  
Critical Points ◽  
Julia Set ◽  
Rational Maps ◽  
Rational Map ◽  
Fatou Component ◽  
Jordan Curves ◽  
Fatou Components

AbstractWe prove: If f(z) is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of f is a Jordan curve. If f(z) is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps f with a Fatou component ω satisfying f(ω) = ω and f|∂ω not topologically conjugate to the dynamics of any polynomial on its Julia set.

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.

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.

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.

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)) = ∞.

-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.

scholarly journals Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles

2010 ◽  
Vol 30 (6) ◽  
pp. 1869-1902 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Jordan Curve ◽  
Riemann Sphere ◽  
Julia Set ◽  
Julia Sets ◽  
John Domain ◽  
Unbounded Component ◽  
Random Dynamics ◽  
Jordan Curves ◽  
Polynomial Maps

AbstractWe investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere $\CCI $) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence $\gamma \in \Gamma ^{\NN }$, the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of $\CCI {\setminus } J_{\gamma }$ is a John domain, and the bounded component of $\CC {\setminus } J_{\gamma }$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

scholarly journals A family of rational maps with buried Julia components

2014 ◽  
Vol 35 (6) ◽  
pp. 1846-1879 ◽  
Author(s):  
SÉBASTIEN GODILLON
Keyword(s):  
Explicit Formula ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Jordan Curves ◽  
Polynomial Map

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However, all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least five. This paper introduces a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover, this family contains some rational maps of degree three with explicit formula that answers a question McMullen raised.

scholarly journals Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps

2007 ◽  
Vol 59 (2) ◽  
pp. 311-331 ◽  
Author(s):  
Hans Christianson
Keyword(s):  
Dynamical System ◽  
Lower Bound ◽  
Zeta Function ◽  
Julia Set ◽  
Upper Bounds ◽  
Rational Maps ◽  
Rational Map ◽  
Numerical Work ◽  
Number Of Zeros ◽  
Ruelle Zeta Function

AbstractThis paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp(CK|s|δ) in strips | Re s| ≤ K, where δ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott–Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions {| Re s| ≤ | Im s|α} is given, followed by weaker lower bound estimates in strips {Re s > –C, | Ims| ≤ r}, and logarithmic neighbourhoods {| Re s| ≤ ρlog | Ims|}. Recent numerical work of Strain–Zworski suggests the upper bounds in strips are optimal.

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