Geometric limit of Julia set of a family of rational functions with odd degree

2021 ◽  
pp. 1-0
Author(s):  
A. M. Alves ◽  
B. P. Silva e Silva ◽  
M. Salarinoghabi
Keyword(s):  
Julia Set ◽  
Rational Functions ◽  
Geometric Limit ◽  
Odd Degree

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.

On the transfer operator for rational functions on the Riemann sphere

1996 ◽  
Vol 16 (2) ◽  
pp. 255-266 ◽  
Author(s):  
Manfred Denker ◽  
Feliks Przytycki ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Transfer Operator ◽  
Rational Functions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Correlation Integral ◽  
Hölder Continuous Functions ◽  
Conformal Measure

AbstractLet T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

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.

scholarly journals Non-archimedean connected Julia sets with branching

2015 ◽  
Vol 37 (1) ◽  
pp. 59-78
Author(s):  
DVIJ BAJPAI ◽  
ROBERT L. BENEDETTO ◽  
RUQIAN CHEN ◽  
EDWARD KIM ◽  
OWEN MARSCHALL ◽  
...  
Keyword(s):  
Topological Entropy ◽  
Line Segment ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Dynamical Property ◽  
Projective Line

We construct the first examples of rational functions defined over a non-archimedean field with a certain dynamical property: the Julia set in the Berkovich projective line is connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we give an example for which the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.

scholarly journals Waring’s Problem for Rational Functions in One Variable

2020 ◽  
Vol 71 (2) ◽  
pp. 439-449
Author(s):  
Bo-Hae Im ◽  
Michael Larsen
Keyword(s):  
Rational Function ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Laurent Polynomial ◽  
Necessary And Sufficient Conditions ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Odd Degree ◽  
Necessary And Sufficient ◽  
Polynomial Of Odd Degree

Abstract Let $f\in{\mathbb{Q}}(x)$ be a non-constant rational function. We consider ‘Waring’s problem for $f(x)$,’ i.e., whether every element of ${\mathbb{Q}}$ can be written as a bounded sum of elements of $\{f(a)\mid a\in{\mathbb{Q}}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring’s problem. We also consider the ‘easier Waring’s problem’: whether every element of ${\mathbb{Q}}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in{\mathbb{Q}}\}$.

On the residual Julia sets of rational functions

1997 ◽  
Vol 17 (1) ◽  
pp. 205-210 ◽  
Author(s):  
SHUNSUKE MOROSAWA
Keyword(s):  
Kleinian Group ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Limit Set ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON ℝ2

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950099 ◽  
Author(s):  
ROBERTO DE LEO
Keyword(s):  
Julia Set ◽  
Rational Functions ◽  
Real Polynomial ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Open Set ◽  
Complex Roots ◽  
Wandering Domain ◽  
Set Of Points ◽  
Real Polynomial Maps

We collect from several sources some of the most important results on the forward and backward limits of points under real and complex rational functions, and in particular real and complex Newton maps, in one variable and we provide numerical evidence that the dynamics of Newton maps [Formula: see text] associated to real polynomial maps [Formula: see text] with no complex roots has a complexity comparable with that of complex Newton maps in one variable. In particular such a map [Formula: see text] has no wandering domain, almost every point under [Formula: see text] is asymptotic to a fixed point and there is some non-empty open set of points whose [Formula: see text]-limit equals the set of non-regular points of the Julia set of [Formula: see text]. The first two points were proved by B. Barna in the real one-dimensional case.

Rational functions with no recurrent critical points

1994 ◽  
Vol 14 (2) ◽  
pp. 391-414 ◽  
Author(s):  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Critical Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Necessary And Sufficient

AbstractLet h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

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