On periods of Herman rings and relevant poles

2020/07/14 by Ghora, Subhasis, Nayak, Tarakanta
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.07036

Abstract

Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called H-relevant 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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• On the formulas of meromorphic functions with periodic Herman rings
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• A priori bounds and degeneration of Herman rings with bounded type rotation number
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