Herman rings with small periods and omitted values

Chakra, Tarun Kumar; Chakraborty, Gorachand; Nayak, Tarakanta

doi:10.1016/s0252-9602(18)30858-0

Cited by 9 publications

(10 citation statements)

References 7 publications

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“…If the length of the basic chain is equal to the period of the Herman ring then it follows from the definition of the basic chain that there is only one H 1 -relevant pole, i.e., the total number of distinct poles surrounded by any of the Herman rings of the cycle is 1. However the number of H-relevant poles of every Herman ring of a function with an omitted value is at least two by Lemma 2.11, [6]. Hence the length of the basic chain corresponding to a p-cycle of Herman rings is at most p − 1.…”

Section: Preliminary Resultsmentioning

confidence: 93%

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Rotation domains and Stable Baker omitted value

Ghora¹,

Nayak²

2021

Preprint

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A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of f −1 (D) is bounded. Assuming all the iterates f n are analytic in a neighborhood of its bov, this article proves that the number of Herman rings of a particular period is finite and every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of f n on W is either a parabolic periodic point or in the ω-limit set of recurrent critical points. Using the same ideas, the boundary of rotation domains are shown to be in the ω-limit set of recurrent critical points.

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Section: Preliminary Resultsmentioning

confidence: 93%

“…3. Since the innermost ring H 1 with respect to the bov never surrounds a pole by Remark 2.10 of [6], the length of the basic chain is at least two.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Rotation domains and Stable Baker omitted value

Ghora¹,

Nayak²

2021

Preprint

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“…Bergweiler [11] provides criteria for the escaping set and the Julia set of an entire function to have positive Lebesgue measure. For transcendental meromorphic functions, the dynamics of a family shows in [58] and Herman rings with small periods as well omitted values determine in [16]. The dynamical properties of the extraneous fixed points characterize in [26] with respect to their stability, basins of attraction, cycles, etc.…”

Section: Different Approaches To Studying Dynamics Of One Variable Co...mentioning

confidence: 99%

On Some Advances in Dynamics of One Variable Complex Functions

Sajid¹

2022

Int. J. of Appl. Math.

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In this article, we present some recent advances in the dynamics of one variable complex functions which are especially associated to the Julia sets, the Fatou sets, the Sigel disks, the Herman rings, fixed points as well as the Mandelbrot sets. For the sake of interest, we consider mainly research works which have been done from 2015 to 2019. To achieve our purpose, some interesting and selected themes are considered to show some recent advances in the dynamics of one variable complex functions.

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“…Examples of functions which has no Herman ring are also provided in [6]. In view of all these, following conjecture can be made.…”

Section: Introductionmentioning

confidence: 99%

“…We say a set is surrounded by a Herman ring H if the set is contained in the bounded component of the complement of H. The locations of the omitted value(s) and poles surrounded by Herman rings have also been key to a number of useful observations. Later, these observations are used to show that there cannot be more than one p-cycles of Herman rings for p = 3, 4 [6]. These ideas are developed and used in this article to prove a lower bound for periods of Herman rings and non-existence of the same under certain situation.…”

Section: Introductionmentioning

confidence: 99%

On periods of Herman rings and relevant poles

Ghora¹,

Nayak²

2020

Preprint

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Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called Hrelevant for a Herman ring H of such a function f if it is surrounded by some Herman ring of the cycle containing H. In this article, a lower bound on the period p of a Herman ring H is found in terms of the number of H-relevant poles, say h. More precisely, it is shown that p ≥ h(h+1) 2 whenever f j (H), for some j, surrounds a pole as well as the set of all omitted values of f . It is proved that p ≥ h(h+3) 2 in the other situation. Sufficient conditions are found under which equalities hold.It is also proved that if an omitted value is contained in the closure of an invariant or a two periodic Fatou component then the function does not have any Herman ring.

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Herman rings with small periods and omitted values

Cited by 9 publications

References 7 publications

Rotation domains and Stable Baker omitted value

Rotation domains and Stable Baker omitted value

On Some Advances in Dynamics of One Variable Complex Functions

On periods of Herman rings and relevant poles

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