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Some result on escaping set of an entire function
Abstract
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}.
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We show that an invariant Fatou component of a hyperbolic transcendental
entire function is a bounded Jordan domain (in fact, a quasidisc) if and only
if it contains only finitely many critical points and no asymptotic curves. We
use this theorem to prove criteria for the boundedness of Fatou components and
local connectivity of Julia sets for hyperbolic entire functions, and give
examples that demonstrate that our results are optimal. A particularly strong
dichotomy is obtained in the case of a function with precisely two critical
values.
We show that for many complex parameters a, the set of points that converge
to infinity under iteration of the exponential map f(z)=e^z+a is connected.
This includes all parameters for which the singular value escapes to infinity
under iteration.
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.
Let f be an entire function of finite order whose set of singular values is bounded or, more generally, a finite composition of such functions. We show that every escaping point of f can be connected to ∞ by a curve in I(f). This provides a positive answer to a question of Eremenko for a large class of entire functions.
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
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.
The points which converge to ∞ under iteration of the maps z↦λexp(z) for λ ∈ C/{0} are investigated. A complete classification of such ‘escaping points’ is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ.
It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of λ.
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.
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
The escaping set of the exponential, Ergodic Theory and Dynamical Systems
- L Rempe
L. Rempe, The escaping set of the exponential, Ergodic Theory and Dynamical Systems, 30 (2010),
595-599.
- W Bergweiler
- N Fagella
- L Rempe
W. Bergweiler, N. Fagella and L. Rempe, Hyperbolic entire functions with bounded Fatou components,
arXiv:math.DS/14040925, (2014).