# Is the set of escaping endpoints for $e^z-2$ completely metrizable?

Let $$f:\mathbb C \to \mathbb C$$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $$J(f)$$, the Julia set of $$f$$, is a uncountable collection of disjoint rays (one-to-one continuous images of $$[0,\infty)$$). It looks something like the figure below. The number $$2$$ is rather arbitrary; the dynamics of $$f$$ are practically identical if we replace $$2$$ with any number greater than $$1$$.

Let $$E(f)$$ be the set of all $$0$$-endpoints of these rays, i.e. the "endpoints" of $$J(f)$$.

Let $$I(f)=\{z\in \mathbb C:f^n(z)\to\infty\}$$. Here $$f^2$$ is the composition $$f\circ f$$, etc.

Let $$\tilde E(f)=I(f)\cap E(f)$$ be the set of escaping endpoints of $$J(f)$$.

It is known that $$E(f)$$ is completely metrizable.

Question. Is $$\tilde E(f)$$ completely metrizable?

We may independently consider the set of escaping points. It is easy to show $$I(f)$$ is an $$F_{\sigma\delta}$$-subset of the plane using only continuity of $$f$$.

Question. Is $$I(f)$$ completely metrizable?

Alhabib, Nada; Rempe-Gillen, Lasse, Escaping endpoints explode, Comput. Methods Funct. Theory 17, No. 1, 65-100 (2017). ZBL1381.37051. for further information.

• A subset of a complete metric space is completely metrizable if and only if it is a $G_\delta$, so the questions are equivalent to ask whether these sets are $G_\delta$ Aug 4 '19 at 16:53

$$\tilde E(f)$$ and $$I(f)$$ are first category, so the answer to each question is NO. https://doi.org/10.1017/etds.2019.111