Iterating the minimum modulus

Abstract: For an entire function $f$ there may or may not exist an $r > 0$ such that the iterated minimum modulus $m^n(r)$ tends to infinity. Here $m(r) = m(r,f) = \min\{ |f(z)| : |z|=r \}$. Focussing mainly on the class of real transcendental entire functions of finite order with only real zeroes, we discuss some results about the existence of an $r > 0$ such that $m^n(r) \to \infty$. This is motivated by the result that, for functions in this class, the existence of such an r implies connectedness of the escaping set $\{ z : f^n(z) \to \infty \}$.

This is joint work with Phil Rippon and Gwyneth Stallard.

complex variablesdynamical systems

Audience: researchers in the topic

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