Given \(q\geq 1\), let \(\mathcal{E}_q\) be the family of transcendental entire functions of the form \(f(z)=\sum_{j=0}^{q-1} a_k\exp(\omega_q^k z)\), where \(\omega_q=\exp(2\pi i/q)\) and \(a_0,\ldots,a_{q-1}\neq 0\). This paper studies the dynamics of functions in \(\mathcal{E}_q\) for \(q\geq 2\), generalizing and adapting what is known for \(\mathcal{E}_1\) (the exponential family) and \(\mathcal{E}_2\) (the cosine family). This is interesting because while \(\mathcal{E}_1\cup\mathcal{E}_2\) is a subset of the Emerenko-Lyubich class of entire functions with bounded singular set; if \(q\geq 3\) then the singular set of an element of \(\mathcal{E}_q\) is never bounded.
The author proves first of all that if \(f\in\mathcal{E}_q\) with \(q\geq 2\) then \(J(f)\cap A(f)\) and \(J(f)\cap I(f)\) have positive area, where \(J(f)\) is the Julia set of \(f\), \(I(f)\) is its escaping set and \(A(f)\) is its fast escaping set. Then he proves that if \(f\in\mathcal{E}_q\) with \(q\geq 3\) then \(A(f)\), \(I(f)\) and \(J(f)\) are spider’s webs with positive area.