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 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.

LIMITING JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

2008 ◽  
Vol 18 (10) ◽  
pp. 3175-3181 ◽  
Author(s):  
MARK MORABITO ◽  
ROBERT L. DEVANEY
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Julia Set ◽  
Singularly Perturbed ◽  
Parameter Plane ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Open Set ◽  
The Family

In this paper, we consider the family of rational maps given by [Formula: see text] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

Parabolic-like mappings

2014 ◽  
Vol 35 (7) ◽  
pp. 2171-2197 ◽  
Author(s):  
LUNA LOMONACO
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
The Family ◽  
Dynamical Properties ◽  
Image Position ◽  
Parabolic Fixed Point

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family $$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$ a unique such member if the filled Julia set is connected.

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)) = ∞.

Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products

2001 ◽  
Vol 21 (2) ◽  
pp. 563-603 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Hausdorff Dimension ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Empty Interior ◽  
Skew Products ◽  
Wandering Domain ◽  
Upper Estimate ◽  
Interior Points

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to zero or an upper estimate of the Hausdorff dimension of the Julia set.

PARABOLIC RATIONAL MAPS

2001 ◽  
Vol 63 (3) ◽  
pp. 673-689 ◽  
Author(s):  
NICOLAI HAYDN ◽  
STEFANO ISOLA
Keyword(s):  
Asymptotic Distribution ◽  
Spectral Properties ◽  
Julia Set ◽  
Markov Partition ◽  
Escape Rate ◽  
Rational Maps ◽  
Compact Set ◽  
Dynamical Properties

The paper studies the dynamics of rational maps with indifferent parabolic points by comparing their dynamical properties with those of their ‘jump transformation’ which is uniformly expanding on a non-compact set with infinite Markov partition. It establishes the spectral properties of a two-variables operator-valued function associated to the jump transformation and exploits their dynamical relevance to allow the analytic properties of the pressure, the escape rate from a neighbourhood of the Julia set and the asymptotic distribution of pre-images to be studied.

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.

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