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On dynamics of semiconjugated entire functions
Abstract
Let g and h be transcendental entire functions and let f be a
continuous map of the complex plane into itself with Then
g and h are said to be semiconjugated by f and f is called a
semiconjugacy. We consider the dynamical properties of semiconjugated
transcendental entire functions g and h and provide several conditions
under which the semiconjugacy f carries Fatou set of one entire function into
the Fatou set of other entire function appearing in the semiconjugation. We
have also shown that under certain condition on the growth of entire functions
appearing in the semiconjugation, the number of asymptotic values of the
derivative of composition of the entire functions is bounded.
... Recall that if g and h are transcendental entire functions and f is a continuous map of the complex plane into itself with f @BULLET g = h @BULLET f, then g and h are said to be semiconjugated by f and f is called a semiconjugacy [5]. In [11], the first author considered the dynamics of semiconjugated entire functions and provided several conditions under which the semiconjugacy carries Fatou set of one entire function into Fatou set of other entire function appearing in the semiconjugation. Furthermore, it was shown that under certain conditions on the growth of entire functions appearing in the semiconjugation, the set of asymptotic values of the derivative of composition of the entire functions is bounded. ...
We study some relation between escaping sets of two permutable entire functions. In addition, we investigate the dynamical properties of the map $f(z)=z+1+e^{-z}.
We consider the dynamical properties of transcendental entire functions and
their compositions. We give several conditions under which Fatou set of a
transcendental entire function f coincide with that of where g
is another transcendental entire function. We also prove some result giving
relationship between singular values of transcendental entire functions and
their compositions.
We define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of Σ share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets.
Let f and g be two transcendental entire functions. In this paper, mainly by using Iversen's theorem on the singularities, we studied the dynamics of composite functions. We have proved that the Fatou sets of f o g and g o f have the same dynamical properties.
. In this paper, it is shown that finite limit functions of iterates of entire functions in wandering domains are limit points of the forward orbits of the finite singularities of the inverse function. From this the absence of wandering domains for some classes of entire functions is deduced. 1. Introduction and results Let f be a nonlinear entire function. The Fatou set F is the subset of the complex plane where the iterates f n of f form a normal family. The complement of F is called the Julia set and denoted by J . If U is a component of F , then f n (U) is contained in some component of F which we denote by Un . If Un 6= Um for all n 6= m , then U is called wandering. Otherwise U is called preperiodic. In particular, if Un = U for some n , then U is called periodic. Sullivan [25, 26] proved that rational functions do not have wandering domains. Transcendental entire functions, however, may have wandering domains, cf. [2, 3, 5, 10, 11, 17, 26]. On the other hand, certain cla...
It is proved that if U and V are connected components of the Fatou set of an entire function f and if f(∪)⊂ V, then V\f(∪) contains at most one point.
Let g and h be transcendental entire functions and let f be continuous and open map on the complex plane. Let (g∘f)(z)=(f∘h)(z) for every z∈ℂ. Then g and h are said to be semiconjugated by f. In this paper we obtain several results on semiconjugated entire functions. For instance we show that for the above functions if there exist constants α>0 and r 0 >0 such that |f(z)|>α for all z∈(|z|>r 0 ) then f(F(h))⊂F(g), where F(·) denotes the Fatou set of the function.
Let f and h be two entire functions and let g : C → C be a non- constant continuous function such that f ◦ g = g ◦ h. Then we say that f and h are semiconjugated by g and we call g a semiconjugacy. In this paper, we have shown that under some conditions on the growth of semiconjugacy, the growth of semiconjugated entire functions can be controlled. We also have proved a result on asymptotic values of derivative of composition of entire functions. Mathematics Subject Classification: 30D35, 30D05, 37F10
Fonctions entieres a domaines multiplement connexes de normalite. Ensembles de Fatou-Julia pour des fonctions qui commuttent. Exemples de type d'Herman. Classes de fonctions entieres sans domaines d'errance
It is proved that if U and V are connected components of the Fatou set of an entire function f and if , then contains at most one point.
If f(z) is meromorphic in C and N1 and N2 are components of the Fatou set of f(z) such that f(z): N1 → N2, it is shown that D = N2 \ f(N1) is a set which contains at most two points. If f(z) is entire then D contains at most one point. Examples show that these results are sharp and also that the points of D are in general neither Picard exceptional nor Nevanlinna deficient.
Let f and h be transcendental entire functions and let
g be a continuous and open map of the complex plane into itself with
g[circ B: composite function (small circle)]f=h[circ B: composite function (small circle)]g. We show that if f satisfies
a certain condition, which holds, in particular, if f has no wandering domains, then
g−1(J(h))=J(f). Here
J(·) denotes the Julia set of a function. We conclude that if f
has no wandering domains, then h has no wandering domains. Further, we show that
for given transcendental entire functions f and h, there are only
countably many entire functions g such that
g[circ B: composite function (small circle)]f=h[circ B: composite function (small circle)]g.
For functions meromorphic in the plane, apart from an exceptional case, the Julia set J is the closure of the set of all preimages of poles. The repelling periodic cycles are dense in J. In contrast with the case of transcendental entire functions, J may be a subset of a straight line and general classes of functions for which this is the case can be determined. J may also lie on a quasicircle through infinity which is not a straight line.
Letf andg be nonlinear entire functions. The relations between the dynamics off⊗g andg⊗f are discussed. Denote byℐ (·) andF(·) the Julia and Fatou sets. It is proved that ifz∈C, thenz∈ℐ8464 (f⊗g) if and only ifg(z)∈ℐ8464 (g⊗f); ifU is a component ofF(f○g) andV is the component ofF(g○g) that containsg(U), thenU is wandering if and only ifV is wandering; ifU is periodic, then so isV and moreover,V is of the same type according to the classification of periodic components asU. These results are used to show that certain new classes of entire functions do not have wandering domains.
In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying
and , for some transcendental entire function h , rational function and a function , which is analytic in the range of h , then . Then as an application of this result, we show that if , where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or , where p, q are nonconstant polynomials, such that f(g)=g(f) for a nonconstant entire function g , then J(f)=J(g) .
Let f and g be two permutable transcendental entire functions. Assume that f has the form f(z) = p(z) + p1 (z)eq1(z) + p2 (z) eq2(z) We shall investigate the dynamical properties of f and g and show that they have the same Julia sets and Fatou sets, i.e., J(f) = J(g). This relates to an open question due to Baker.
Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ?, then every asymptotic value of f, except at most 2? of them, is a limit point of critical values of f.
We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f'fn with n = 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions
This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains). Comment: 38 pages. Abstract added in migration. See http://analysis.math.uni-kiel.de/bergweiler/ for recent comments and corrections
Iterates of meromorphic functions I, Ergodic Theory and Dynamical Systems
- I N Baker
- J Kotus
- Lu Yinian
I. N. Baker, J. Kotus and Lu. Yinian, Iterates of meromorphic functions I, Ergodic Theory and
Dynamical Systems, 11 (1991), 241-248.
- D Kumar
- G Datt
- S Kumar
D. Kumar, G. Datt and S. Kumar, Dynamics of composite entire functions, arXiv:math.DS/12075930,
(2013), submitted for publication.