Herman Rings of Elliptic Functions
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Herman Rings of Elliptic Functions Mónica Moreno Rocha1 Received: 14 January 2020 / Revised: 26 August 2020 / Accepted: 24 October 2020 / Published online: 11 November 2020 © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
Abstract It has been shown by Hawkins and Koss that over any given lattice, the Weierstrass ℘ function does not exhibit cycles of Herman rings. We show that, regardless of the lattice, any elliptic function of order two cannot have cycles of Herman rings. Through quasiconformal surgery, we obtain the existence of elliptic functions of order at least three with an invariant Herman ring. Finally, if an elliptic function has order o ≥ 2, then we show there can be at most o − 2 invariant Herman rings. Keywords Elliptic functions · Herman rings · Quasiconformal surgery Mathematics Subject Classification 37F10 · 37F50 · 30D05
1 Introduction Given a transcendental meromorphic function f : C → C with a unique essential singularity at infinity, we consider the dynamical system determined by its iterates. The Fatou set of f , denoted by F( f ), is defined as the set of points z ∈ C that have an open neighborhood U = U (z) where the family of iterates { f n |U } is well defined and form a normal family. The Julia set, J ( f ) equals the complement of F( f ) with respect to C. In particular, J ( f ) = n≥0 f −n (∞). If f is an elliptic function with respect to a lattice (that is, a meromorphic function which is -periodic), its order, denoted by o f , is defined as the number of poles of f in a given period parallelogram, counting multiplicity. Furthermore, f takes each value z ∈ C exactly o f times. Hence, f has no omitted or asymptotic values, and
The author was supported by Consejo Nacional de Ciencia y Tecnología (CONACyT) through Complementary Support for Consolidation of Research Groups, call 2015.
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Mónica Moreno Rocha [email protected] Centro de Investigación en Matemáticas (CIMAT), Callejon Jalisco sn, 36023 Guanajuato, GTO, Mexico
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its singular set consists of a finite number of critical values. One can conclude that f neither exhibits wandering nor Baker domains, and the classification of periodic Fatou components of any elliptic function reduces to (super)attracting, parabolic and rotation domains (Siegel disks or Herman rings). In general, it is a difficult problem to determine the existence of Herman rings since, in contrast to other Fatou components, a Herman ring is not associated with a periodic orbit. On the other hand, each cycle of Herman rings is closely related to the post-critical set and poles of the function. In the context of elliptic functions, Hawkins and Koss showed in [8] that over any given lattice , the Weierstrass ℘-function has no cycle of Herman rings. The proof relies on two facts: the set of poles for ℘ coincides with the set of periods and ℘ is an even function. In the particular case when H is a positively invariant Herman ring under ℘ , one can assume that the origin lies inside its bounded compleme
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