On Fatou sets containing Baker omitted value

Ghora, Subhasis; Nayak, Tarakanta; Sahoo, Satyajit

doi:10.48550/arxiv.2008.09797

Cited by 1 publication

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“…We put Theorem 2.1 and Lemma 2.3(1) of [9] together as a lemma that exhibits the influence of the bov on all other singular values of the function.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…The function f has only one asymptotic value and that is the bov. Some useful observations on the dynamics of functions with stable bov are made in Lemma 2.4 and Theorem 3.3 of [9]. We collect them here.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Under the assumption of the above theorem, U is not wandering giving that its grand orbit cannot contain any wandering domain. Further, every other possible wandering domain of f ∈ M S is simply connected by Theorem 3.1(1) [9]. This gives that the boundary of wandering domains is connected.…”

Section: Introductionmentioning

confidence: 85%

“…Then B is a forward invariant compact subset of the union of F (f ) and {z : z is a parabolic periodic point of f }. It is important to note here that f ∈ M S has no Baker domain by Theorem 3.4 [9]. Note that C\B is a backward invariant open set containing all the critical points of f belonging to the Julia set.…”

Section: If Two Boundary Components Of Two Herman Rings Intersect J Thenmentioning

confidence: 99%

“…A Fatou component is a maximal connected subset of the Fatou set. It is very important to note that for all functions in M S , all but one Fatou component are bounded (See Lemma 2.4 (2) [9] or Lemma 2.5 in the next section). This fact has been crucial in [9] to determine the connectivity of all the Fatou components.…”

Section: Introductionmentioning

confidence: 99%

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Rotation domains and Stable Baker omitted value

Ghora¹,

Nayak²

2021

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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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“…We put Theorem 2.1 and Lemma 2.3(1) of [9] together as a lemma that exhibits the influence of the bov on all other singular values of the function.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 85%

Section: If Two Boundary Components Of Two Herman Rings Intersect J Thenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations