Herman rings of elliptic functions

Mónica Moreno Rocha (Centro de Investigación en Matemáticas, Mexico)

19-Jan-2021, 14:00-15:00 (3 years ago)

Abstract: Consider the family of iterates of a rational or transcendental meromorphic function $f$. A component of normality that is invariant under some $n$-iterate of $f$ is called a Herman ring if over such a component, $f^n$ is conformally conjugate to an irrational rotation acting on an annulus of finite conformal modulus. In that case, the positive iterates of the Herman ring form a cycle. Showing the existence of cycles of Herman rings for meromorphic functions is not an easy task, and when they exist, it is natural to ask oneself if an upper bound for the number of cycles is achievable.

In the late 1980s Shishikura introduced the theory of quasiconformal surgery to construct examples of rational maps with cycles of Herman rings while also showing that a rational map of degree d has at most d-2 cycles (thus, rational maps of degree 2 cannot have Herman rings). In the case of elliptic functions, Hawkins & Koss showed in 2004 that the Weierstrass P function, defined over any given lattice, cannot have cycles of Herman rings. This result motivated the question of the existence of Herman rings for elliptic functions in terms of their order. In this talk, I will present recent results obtained through the implementation of Shishikura’s surgery techniques to the elliptic case. First, we’ll see that Herman rings can be realizable by elliptic functions of order at least 3, and in particular, order 2 elliptic functions cannot have cycles of Herman rings. Then, I will present an upper bound for the number of invariant Herman rings in terms of the order of the elliptic function and show how to refine that bound using the multiplicity of poles.

complex variablesdynamical systems

Audience: researchers in the topic


CAvid: Complex Analysis video seminar

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