Constructing entire functions of small order - motivated by complex dynamics

Abstract: In 1989, Eremenko conjectured that for any transcendental entire function the escaping set $I(f) = \{z:f^n(z)\to\infty \text{ as } n\to\infty\}$ has no bounded components -- despite much work this conjecture is still open.

For real entire functions $f$ of finite order with only real zeros, we have shown that Eremenko's conjecture holds if there exists $r>0$ such that the iterated minimum modulus $m^n(r)\to\infty$ as $n\to\infty$. Here $m(r)=\min_{|z|=r}|f(z)|$.

We discuss examples of families of entire functions of small order for which this iterated minimum modulus condition holds, and construct examples of functions of small order for which it does not hold, including examples based on a new development of a method due to Kjellberg.

(Joint work with Dan Nicks and Gwyneth Stallard.)

complex variablesdynamical systems

Audience: researchers in the topic

Comments: Please e-mail Rod Halburd (r.halburd@ucl.ac.uk) for the Zoom link. Please let him know if you would like to receive weekly announcements about CAvid (the Complex Analysis video seminar series).

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CAvid: Complex Analysis video seminar

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