A bouquet of pseudo-arcs

TL;DR

This paper constructs a simpler, explicit transcendental entire function with a Julia set forming a bouquet of pseudo-arcs, advancing understanding of complex dynamics and fractal structures.

Contribution

It provides a more straightforward, explicit construction of a transcendental entire function with a Julia set composed of pseudo-arcs, improving on previous complex examples.

Findings

The constructed function has lower order 1/2.

The Julia set forms a bouquet of pseudo-arcs.

The construction simplifies previous complex examples.

Abstract

We prove the existence of a transcendental entire function whose Julia set is a "bouquet of pseudo-arcs". More precisely, the union of the Julia set with infinity is an uncountable union of pseudo-arcs, which are pairwise disjoint except at infinity. The existence of such a function follows from a more general result of the second author, but our construction is considerably simpler and more explicit. In particular, the function we construct can be chosen to have lower order $1/2$, while the lower order in the previously known example is infinite.