To be confirmed

In 1926, Fatou asked whether, for a general class of entire
functions, the escaping set (points that tend to infinity under
repeated application of the function) contains curves to
infinity. This was answered by RottenfuBer, Ruckert, Rempe
and Schleicher in 2011, who showed that, for functions with
bounded singular set and finite order of growth, the entire
escaping set consists of such curves, called "hairs". They also
prove the existence of a function of "small" infinite order
where the Julia set contains no hairs at all.
There is, however, a gap between the two growth conditions
(finite order and the growth of the counterexample). The goal
of the project is to investigate whether the condition of finite
order is optimal. There are a number of related questions that
can also be investigated, in particular regarding so-called
"Cantor bouquets".