scholarly journals DEGENERATIONS OF COMPLEX DYNAMICAL SYSTEMS

2014 ◽  
Vol 2 ◽  
Author(s):  
LAURA DE MARCO ◽  
XANDER FABER
Keyword(s):  
Equilibrium Measure ◽  
Riemann Sphere ◽  
Projective Line ◽  
Weak Limit ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Uniqueness Of Solutions ◽  
The Family ◽  
A New Technique ◽  
Measures Of Maximal Entropy

AbstractWe show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere ${\hat{{\mathbb{C}}}} $ must be a countable sum of atoms. For a one-parameter family $f_t$ of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on $\hat{{\mathbb{C}}}$ as the family degenerates. The family $f_t$ may be viewed as a single rational function on the Berkovich projective line $\mathbf{P}^1_{\mathbb{L}}$ over the completion of the field of formal Puiseux series in $t$, and the limiting measure on $\hat{{\mathbb{C}}}$ is the ‘residual measure’ associated with the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$.

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).

REMARKS ON LARGE DEVIATION FOR RATIONAL MAPS ON THE RIEMANN SPHERE

2007 ◽  
Vol 07 (03) ◽  
pp. 357-363 ◽  
Author(s):  
HONGQIANG XIA ◽  
XINCHU FU
Keyword(s):  
Large Deviation ◽  
Riemann Sphere ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Measure Of Maximal Entropy

We consider rational maps of degree d ≥ 2 on the Riemann sphere and obtain large deviation results for Hölder observables under the measure of maximal entropy.

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.

Equilibrium measures for rational maps

1986 ◽  
Vol 6 (3) ◽  
pp. 393-399 ◽  
Author(s):  
Artur Oscar Lopes
Keyword(s):  
Equilibrium Measure ◽  
Julia Set ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Equilibrium Measures ◽  
Polynomial Map ◽  
Two Measures ◽  
Measure Of Maximal Entropy

AbstractFor a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

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.

The Family of Goals-Achievement Matrix Methods: Respectable Enough for Citizen Participation in Planning?

1981 ◽  
Vol 13 (9) ◽  
pp. 1151-1161 ◽  
Author(s):  
T Sager
Keyword(s):  
Citizen Participation ◽  
Production Model ◽  
Spatial Equilibrium ◽  
Land Rent ◽  
Matrix Methods ◽  
Differential Fertility ◽  
The Family ◽  
Planning Processes ◽  
A New Technique ◽  
Evaluation Techniques

This article examines the problems of location and land rent within the framework of a linear and multisector production model in a radioconcentric space. In the case of a single product, the order of the differential fertility and location rent is examined. This order is neither given nor natural, but depends especially on the distribution. Formulating a spatial equilibrium with n goods where several qualities of soil exist raises a number of difficulties. When several techniques are used to manufacture a product then it is shown how different techniques are located in relation to the centre according to prices and the distribution variables. The manner in which a new technique is introduced and spreads in space is also examined. The predominant method used for evaluation in British structure planning compares strategies in terms of their achievement of particular criteria derived from community objectives. A recent survey shows that of the traditional evaluation techniques only modified goals-achievement matrix (GAM) methods are widely used in structure planning. This approach is also applied to local and project oriented planning processes where citizen participation seems even more necessary. The goals-achievement matrix was not presented by its originator—M Hill—as an entirely unambiguous method. It can be interpreted as a set of significantly diverging variants which are used here for three purposes: to show that economists' critique of GAM is too general, to clarify the connections between GAM and other well-known evaluation methods, and to discuss how GAM could best be structured for use in local participatory planning.

scholarly journals Construction of New S-Box Using Action of Quotient of the Modular Group for Multimedia Security

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Imran Shahzad ◽  
Qaiser Mushtaq ◽  
Abdul Razaq
Keyword(s):  
Finite Field ◽  
Modular Group ◽  
Fibonacci Sequence ◽  
Projective Line ◽  
Multimedia Security ◽  
New Technique ◽  
Substitution Box ◽  
Nonlinear Component ◽  
Coset Diagrams ◽  
A New Technique

Substitution box (S-box) is a vital nonlinear component for the security of cryptographic schemes. In this paper, a new technique which involves coset diagrams for the action of a quotient of the modular group on the projective line over the finite field is proposed for construction of an S-box. It is constructed by selecting vertices of the coset diagram in a special manner. A useful transformation involving Fibonacci sequence is also used in selecting the vertices of the coset diagram. Finally, all the analyses to examine the security strength are performed. The outcomes of the analyses are encouraging and show that the generated S-box is highly secure.

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.

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