Baker omitted value

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

doi:10.1080/17476933.2016.1174216

Cited by 7 publications

(20 citation statements)

References 9 publications

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“…A point b ∈ C is called an omitted value of f if f (z) = b for any z ∈ C. A special type of omitted value, introduced by Chakra et al in [5] is the concern of this article. The set of singular values is the closure of the union of the critical values and the asymptotic values.…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 99%

“…Thus, the investigation of dynamics of functions with bov is a new direction in transcendental dynamics. This has been initiated in [5].…”

Section: Introductionmentioning

confidence: 99%

Rotation domains and Stable Baker omitted value

Ghora¹,

Nayak²

2021

Preprint

Self Cite

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“…Thus the study of dynamics of functions with bov is essentially a new direction. It has been initiated in [7].…”

Section: Introductionmentioning

confidence: 99%

“…The amount of influence that a particular type of singular value can have on the dynamics seems to be worth knowing, especially when there are infinitely many singular values. The Baker omitted values are a special type of singular values and some aspects of their importance are explored in [7]. A bov is called stable if it is in the Fatou set of the function.…”

Section: Introductionmentioning

confidence: 99%

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On Fatou Sets Containing Baker Omitted Value

2021

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An omitted value of a transcendental meromorphic function f is called a Baker omitted value, in short bov if there is a disk D centered at the bov such that each component of the boundary of f −1 (D) is bounded. Assuming the bov to be in the Fatou set, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If U is the Fatou component containing the bov then it is proved that a Fatou component U is infinitely connected if and only if it lands on U , i.e. f k (U ) ⊂ U for some k ≥ 1. Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbit does not intersect U is finite, we have shown that the connectivity of each Fatou component belongs to a finite set independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent. It is also proved that, if there is a 2-periodic Baker domain, or a 2-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components.

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Rotation Domains and Stable Baker Omitted Value

Ghora

Nayak

2021

Qual. Theory Dyn. Syst.

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Baker omitted value

Cited by 7 publications

References 9 publications

Rotation domains and Stable Baker omitted value

Rotation domains and Stable Baker omitted value

On Fatou Sets Containing Baker Omitted Value

Rotation Domains and Stable Baker Omitted Value

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