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.

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.

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

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 A generalisation of von Staudt’s theorem on cross-ratios

2019 ◽  
Vol 168 (3) ◽  
pp. 601-612
Author(s):  
YATIR HALEVI ◽  
ITAY KAPLAN
Keyword(s):  
Projective Line ◽  
Transcendence Degree ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractA generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

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 Julia sets of uniformly quasiregular mappings are uniformly perfect

2011 ◽  
Vol 151 (3) ◽  
pp. 541-550 ◽  
Author(s):  
ALASTAIR N. FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Hausdorff Dimension ◽  
Analogous Result ◽  
Julia Set ◽  
Julia Sets ◽  
Quasiregular Mapping ◽  
Higher Dimensions ◽  
Ring Domain ◽  
Rational Map ◽  
Quasiregular Mappings ◽  
Uniformly Perfect

AbstractIt is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

scholarly journals Un théorème de l'application ouverte sur les corps valués algébriquement clos

2012 ◽  
Vol 111 (2) ◽  
pp. 161 ◽  
Author(s):  
Laurent Moret-Bailly
Keyword(s):  
Finite Type ◽  
Rational Points ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Absolute Value ◽  
The Absolute

Let $K$ be an algebraically closed field with a nontrivial absolute value, and let $f:X\to Y$ be a morphism of $K$-schemes of finite type. We show that $f$ is universally open if and only if the induced map on $K$-rational points is open for the topologies deduced from the absolute value. Resumé Soit $K$ un corps algébriquement clos muni d'une valeur absolue non triviale, et soit $f:X\to Y$ un morphisme de $K$-schémas de type fini. On montre que $f$ est universellement ouvert si et seulement si l'application induite sur les points $K$-rationnels est ouverte pour les topologies déduites de la valeur absolue.

scholarly journals Belyi’s theorem in characteristic two

2019 ◽  
Vol 156 (2) ◽  
pp. 325-339 ◽  
Author(s):  
Yusuke Sugiyama ◽  
Seidai Yasuda
Keyword(s):  
Rational Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Obstruction Class

We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.

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