scholarly journals The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of 

2018 ◽  
Vol 154 (7) ◽  
pp. 1441-1472 ◽  
Author(s):  
Dragos Ghioca ◽  
Khoa D. Nguyen ◽  
Hexi Ye
Keyword(s):  
Dynamical System ◽  
Rational Function ◽  
Julia Set ◽  
Small Height ◽  
Bogomolov Conjecture

We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms$\unicode[STIX]{x1D6F7}$of$(\mathbb{P}^{1})^{n}$. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

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.

DIMENSIONS OF JULIA SETS OF HYPERBOLIC MEROMORPHIC FUNCTIONS

2001 ◽  
Vol 33 (6) ◽  
pp. 689-694 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Meromorphic Function ◽  
Box Dimensions

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

scholarly journals Differentiability with respect to parameters of average values in probabilistic contracting dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 599-610 ◽  
Author(s):  
W. Douglas Withers
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Harmonic Function ◽  
Invariant Measure ◽  
Compact Subset ◽  
Julia Set ◽  
Spatial Variable ◽  
External Parameter ◽  
Average Value ◽  
Complex Functions

AbstractWe consider a dynamical system consisting of a compact subset of RN or CN with several contracting maps chosen with prescribed probabilities, which may depend on position. We show that if the maps and the probabilities are Cl+α functions of the spatial variable and an external parameter, then the average value of a Cl+α function is a differentiate function of the parameter. One implication of this theorem is that for certain families of complex functions dependent on a parameter the reciprocal of the dimension of an invariant measure on the Julia set is a harmonic function of the parameter.

Application of the extension exponential rational function method for higher-dimensional Broer–Kaup–Kupershmidt dynamical system

2019 ◽  
Vol 35 (01) ◽  
pp. 1950345 ◽  
Author(s):  
Aly R. Seadawy ◽  
K. El-Rashidy
Keyword(s):  
Dynamical System ◽  
Mathematical Method ◽  
Partial Differential Equations ◽  
Exact Solutions ◽  
Rational Function ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Exponential Functions ◽  
Higher Dimensional ◽  
Exponential Rational Function Method

The extension of exponential rational function method is obtained to construct a series of exact solutions for higher-dimensional Broer–Kaup–Kupershmidt (BKK) dynamical system. New and general solutions are found. The solutions reported in this work are kink solutions, anti-kink solutions and bright solutions. They are expressed in terms of rational exponential functions. A confrontation of our results with the well-known results are done and it comes from this study that the solutions obtained here are new. The mathematical method applied to search for our solutions can be used for other nonlinear partial differential equations. The graphics of the obtained solutions in this paper are shown.

scholarly journals Effective results for points on certain subvarieties of tori

2009 ◽  
Vol 147 (1) ◽  
pp. 69-94 ◽  
Author(s):  
ATTILA BÉRCZES ◽  
KÁLMÁN GYŐRY ◽  
JAN-HENDRIK EVERTSE ◽  
CORENTIN PONTREAU
Keyword(s):  
Number Field ◽  
Finite Rank ◽  
Upper Bounds ◽  
Accurate Description ◽  
Precise Form ◽  
Small Height ◽  
Bogomolov Conjecture ◽  
Special Classes

AbstractThe combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those pointsxof a given subvarietyof$\mathbb{G}_{\bf m}^N(\oQ )=(\oQ^*)^N$, that with respect to the height are “very close” to a given subgroup Γ of finite rank of$\mathbb{G}_{\bf m}^N(\oQ)$. Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.In this paper we prove, for certain special classes of varieties, effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the pointsxunder consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the varietywith very small height.

Dynamical System Visualization and Analysis Via Performance Maps

2004 ◽  
Vol 3 (4) ◽  
pp. 271-287 ◽  
Author(s):  
James J. Alpigini
Keyword(s):  
Dynamical System ◽  
Julia Set ◽  
A Priori ◽  
Julia Sets ◽  
Phase Portraits ◽  
Iterated Function ◽  
Fractal Images ◽  
Visual Impact ◽  
Visualization Techniques ◽  
Dynamical System Models

Visualization techniques are common in the study of chaotic motion. These techniques range from simple time graphs and phase portraits to robust Julia sets, which are familiar to many as ‘fractal images.’ The utility of the Julia sets rests not in their considerable visual impact, but rather, in the color-coded information that they display about the dynamics of an iterated function. In this paper, a paradigm termed the performance map is presented, which is derived from the familiar Julia set. Performance maps are generated automatically for control or other dynamical system models over ranges of system parameters. The resulting visualizations require a minimum of a priori knowledge of the system under evaluation. By the use of color-coding, these images convey a wealth of information to the informed user about dynamic behaviors of a system that may be hidden from all but the expert analyst.

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.

scholarly journals Total disconnectedness of Julia sets of random quadratic polynomials

2021 ◽  
pp. 1-17
Author(s):  
KRZYSZTOF LECH ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical System ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Large Disc ◽  
Fatou Sets ◽  
Autonomous Version ◽  
Composition Of Functions ◽  
Totally Disconnected

Abstract For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

A note on the Julia set of a rational function

1995 ◽  
Vol 118 (3) ◽  
pp. 477-485 ◽  
Author(s):  
S. D. Letherman ◽  
R. M. W. Wood
Keyword(s):  
Rational Function ◽  
Complex Dynamics ◽  
Julia Set ◽  
Julia Sets ◽  
Inverse Image ◽  
Iteration Algorithm ◽  
Finite Sets ◽  
The Subject

The purpose of this note is to present a few facts about the Julia set of a rational function that are well known to the experts in the subject of complex dynamics but whose documented exposition in the literature seems to need a little clarification. For example, under conditions set out in Theorem 1, the Julia set of a rational function can be expressed as the limit of a sequence of finite sets. In particular, for certain choices of a point α, the Julia set is the limit as n increases of the inverse image sets R−n(α). This formulation is widely exploited in the backwards iteration algorithm to produce computer illustrations of Julia sets (see for example Section 5·4 of [14]).

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