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 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 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.

Meromorphic multifunctions and stability of Julia sets

1995 ◽  
Vol 15 (6) ◽  
pp. 1231-1238 ◽  
Author(s):  
Shengjian Wu
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Analytic Family ◽  
Upper Semicontinuous ◽  
The Relationship

AbstractLet Rw(z): W × C∞ → C∞ be an analytic family of rational functions, J(w) the Julia set of Rw and J*(w) the upper semicontinuous regularization of J(w). We shall discuss the relationship between J(w) and J*(w) as well as some related problems.

Julia sets of rational functions are uniformly perfect

1993 ◽  
Vol 113 (3) ◽  
pp. 543-559 ◽  
Author(s):  
A. Hinkkanen
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Uniformly Perfect

AbstractLetfbe a rational function of degree at least two. We shall prove that the Julia setJ(f) offis uniformly perfect. This means that there is a constantc∈(0, 1) depending onfonly such that wheneverz∈J(f) and 0 <r< diamJ(f) thenJ(f) intersects the annulus.

scholarly journals Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

2021 ◽  
pp. 1-18
Author(s):  
YÛSUKE OKUYAMA
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Projective Line ◽  
Closed Field ◽  
Fatou Component ◽  
Algebraically Closed Field ◽  
Absolute Value ◽  
Uniformly Perfect ◽  
Uniform Perfectness

Abstract We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

Fractal analysis and control of the competition model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Mianmian Zhang ◽  
Yongping Zhang
Keyword(s):  
Feedback Control ◽  
Mathematical Models ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Julia Set ◽  
Julia Sets ◽  
Competition Model ◽  
Simulation Results ◽  
Population Competition ◽  
And Control

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

scholarly journals On the Julia sets of rational functions of degree two with two real parameters

1996 ◽  
Vol 26 (2) ◽  
pp. 253-275 ◽  
Author(s):  
Toshio Nakata ◽  
Munetaka Nakamura
Keyword(s):  
Julia Sets ◽  
Rational Functions

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

Sign in / Sign up

Export Citation Format

Share Document