|Location:||MSRI: Online/Virtual, Simons Auditorium|
COMD Stony Brook + MSRI Seminar Series: Meromorphic Functions With A Polar Asymptotic Value
To participate in this seminar, please register HERE.
The work described in this lecture is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. It includes various joint projects with Tao Chen, Nuria Fagella and Yunping Jiang.
Unlike rational maps, transcendental meromorphic functions have an es- sential singularity, and its preimages cannot be iterated indefinitely. It also means the functions have infinite degree, and if there are only finitely many
singular values, there must be asymptotic values — for example, 0 for e^z. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two com-
plex parameters. Like quadratic polynomials and rational maps of degree 2, understanding their parameter spaces is key to understanding families with more asymptotic values.
Starting in the late ’80’s, the exponential family Ea(z) = exp(z)+a, with asymptotic values at 0 and ∞, was studied by Devaney, Goldberg, Rempe and Schleicher, among others, Both the dynamic and parameter spaces exhibit
phenomena not seen for rational maps, for example, “Cantor bouquets”.
The tangent family λ tan z, with asymptotic values ±λi is another example, and, because the asymptotic values are both finite, is more like an “infinite version” of rational maps. Both are one dimensional “slices” of the two dimensional space. Insisting that a fixed point be attractive with a fixed multiplier defines a third example, again more akin to rational maps.
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In this lecture, we look at a fourth family in which one of the asymptotic values is a pole, the “polar asymptotic value” of the title. Although these functions can never be hyperbolic, we will show they exhibit behavior we see in both the tangent and exponential families and interpolate between them.