On the Dynamics of Composition of Transcendental Entire Functions in Angular Region
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RESEARCH ARTICLE
On the Dynamics of Composition of Transcendental Entire Functions in Angular Region Garima Tomar1
Received: 2 February 2017 / Revised: 7 March 2019 / Accepted: 16 May 2019 Ó The National Academy of Sciences, India 2019
Abstract In this paper, we have shown the existence of wandering domains as well as periodic domains with all possible combination under composition of transcendental entire functions in angular region using approximation theory. Keywords Wandering domain Approximation theory Angular region Mathematics Subject Classification 30D05 37F10
1 Introduction Entire functions play a very important role in our day-today life. Iteration of entire functions is used in our real life, like population dynamics. Initially, this tool was used in Newton’s method for finding the roots of a complex analytic function, but modern complex dynamics started with work of Fatou and Julia in early twentieth century. Transcendental entire functions are never infinity in C and are not polynomial, for example exp z. It is known that for entire functions f and g, f g has wandering domain if and only if g f has wandering domain [1]. However, the structure of the Fatou components may have different properties. Singh [2] has shown the existence of component which is wandering for f wandering for g but periodic for f g. Several other similar results were also given in that paper. The structure of periodic domain is well understood & Garima Tomar [email protected] 1
Central University of Rajasthan, Bandarsindri, Kishangarh, Dist., Ajmer, Rajasthan 305817, India
[3], while a lot of work regarding wandering domain has yet to be done. Wandering domains are also very important to study as their existence affects the dynamics of a function, as rational functions do not have wandering domain but transcendental entire function may have. The first example of wandering domain was given by Baker [4]. Since then, several other examples of wandering domain were given by many authors, for instance see [2] and its references. So existence of wandering domains always keeps mathematicians attracting. Let f : C ! C be a transcendental entire function. For n 2 N; f n denotes the nth iteration of f. Thus f n ðzÞ ¼ f ðf n1 ÞðzÞ; where f 0 ðzÞ ¼ z and n ¼ 1; 2; . . . The Fatou set F(f) is defined to be the set of points z 2 C, such thatðf n Þn2N forms a normal family in some neighbourhood of z. The complement of the Fatou set F(f) is called the Julia set J(f). Clearly the Fatou set is open, while the Julia set is closed. For an introduction to the other properties of these sets, one can refer [3, 5]. Also both of these sets are completely invariant. Consequently, if U is a component of F(f), then f(U) is in a component V of F(f). In fact, Vnf ðUÞ contains atmost one point [3]. If Um denotes the component of F(f) containing f m ðUÞ; then if Un \ Um ¼ ; for n 6¼ m then U is called wandering domain. If Um ¼ U and Uk 6¼ U for 1 k\m; then U is called periodic domain of period m. Similar to the iteration theory of tr
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