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Dynamics on Bungee Set of Transcendental entire Functions
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Abstract
In this paper, we have explored some of the basic properties of the Bungee set of a transcendental entire function. We have provided a class of permutable entire functions for which their Bungee sets are equal. Moreover, we have given a class of permutable entire functions for which the escaping set of the composite entire function equals the union of the escaping sets of the two functions. In addition, we provide an important relation between the Bungee set of composite entire function with the Bungee set of individual functions.
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Given two permutable entire functions f and g, we establish vital relationship between escaping sets of entire functions f, g and their composition. We provide some families of transcendental entire functions for which Eremenko’s conjecture holds. In addition, we investigate the dynamical properties of the mapping
We investigate the connectedness properties of the set of
points where the iterates of an entire function f are unbounded. In
particular, we show that is connected whenever iterates of the
minimum modulus of f tend to infinity. For a general transcendental entire
function f , we show that is
always connected and that, if is disconnected, then it has
uncountably many components, infinitely many of which are unbounded.
For a transcendental entire function f, we study the set of points BU(f)
whose iterates under f neither escape to infinity nor are bounded. We give new
results on the connectedness properties of this set and show that, if U is a
Fatou component that meets BU(f), then most boundary points of U (in the sense
of harmonic measure) lie in BU(f). We prove this using a new result concerning
the set of limit points of the iterates of f on the boundary of a wandering
domain. Finally, we give some examples to illustrate our results.
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
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.
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
f k (I(g) ⊂ I(g) for every k ∈ N. In particular, f m k (z 0 ) ∈ I(g) which implies that (g • f ) m k
- Also
Also, using Theorem 3.3, f k (I(g) ⊂ I(g) for every k ∈ N. In particular, f m k (z 0 ) ∈ I(g)
which implies that (g • f ) m k (z 0 ) → ∞ as k → ∞ which is a contradiction.
], showed that for two entire functions f and g, z ∈ F (f • g) if
- Wang Bergweiler
Bergweiler and Wang [6], showed that for two entire functions f and g, z ∈ F (f • g) if
- J Milnor
J. Milnor, Dynamics in One Complex Variable, (third edition), Annals of Math. Studies 160, Princeton U. Press, (2006).
- A P Singh
A. P. Singh,
On bungee sets of composite transcendental entire functions,
arXiv:2006.00208v1[math.CV] (2020), pp. 1-7.