Criniferous entire maps with absorbing Cantor bouquets

Abstract

<p style='text-indent:20px;'>It is known that, for many transcendental entire functions in the Eremenko-Lyubich class <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are <i>criniferous</i>. In this paper, we extend this result to a new class of maps in <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>. Furthermore, we show that if a map belongs to this class, then its Julia set contains a <i>Cantor bouquet</i>; in other words, it is a subset of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{C} $\end{document}</tex-math></inline-formula> ambiently homeomorphic to a straight brush.</p>