BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mónica Moreno Rocha (Centro de Investigación en Matemáticas\, M exico) DTSTART:20210119T140000Z DTEND:20210119T150000Z DTSTAMP:20260422T201544Z UID:CAvid/26 DESCRIPTION:Title: H erman rings of elliptic functions\nby Mónica Moreno Rocha (Centro de Investigación en Matemáticas\, Mexico) as part of CAvid: Complex Analysi s video seminar\n\nLecture held in N/A.\n\nAbstract\nConsider the family o f iterates of a rational or transcendental meromorphic function $f$. A com ponent of normality that is invariant under some $n$-iterate of $f$ is cal led a Herman ring if over such a component\, $f^n$ is conformally conjugat e to an irrational rotation acting on an annulus of finite conformal modul us. 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 i f an upper bound for the number of cycles is achievable.\n\nIn the late 19 80s Shishikura introduced the theory of quasiconformal surgery to construc t 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 m aps of degree 2 cannot have Herman rings). In the case of elliptic functio ns\, Hawkins & Koss showed in 2004 that the Weierstrass P function\, defin ed over any given lattice\, cannot have cycles of Herman rings. This resul t motivated the question of the existence of Herman rings for elliptic fun ctions in terms of their order. In this talk\, I will present recent resul ts obtained through the implementation of Shishikura’s surgery technique s to the elliptic case. First\, we’ll see that Herman rings can be reali zable by elliptic functions of order at least 3\, and in particular\, orde r 2 elliptic functions cannot have cycles of Herman rings. Then\, I will p resent 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 usin g the multiplicity of poles.\n LOCATION:https://researchseminars.org/talk/CAvid/26/ END:VEVENT END:VCALENDAR