Bank-Laine Functions with Real Zeros

Suppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that $$E'(z) = \pm 1$$ E ′ ( z ) = ± 1 whenever $$E(z) = 0$$ E ( z ) = 0 . Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

Connectedness properties of the set where the iterates of an entire function are bounded

We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K(f) is totally disconnected if and only if each component of K(f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is totally disconnected and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.