Points attracting to 0 are dense in $\mathbb C$

Question

I know that the following proposition is true, but at the moment I can't see how to prove it.

• Define $f(z)=e^z-1$ for all $z\in \mathbb C$. Then $A:=\{z\in \mathbb C:f^n(z)\to 0\}$ is dense in $\mathbb C$.

Using an elementary understanding of $f$, one easily sees that $\{z\in \mathbb C:\text{Re}(z)\leq 0\}\subset A$, and that $A$ extends into the right half-plane. But how can we prove it is dense?

Answer 1

This is a corollary of some basic results in dynamics of entire functions. The scheme of the proof is the following:

a) the attraction basin of 0 is not empty (since 0 is a rational neutral fixed point).

b) the only singular value -1 must be attracted to 0 (Fatou's theorem)

c) there are no other cycles of Fatou domains (this uses Fatou's classification of basins of attraction, their relation to singular values, and the No-Wandering-Domains Theorem)

d) Thus Fatou set coincides with the basin of 0.

e) The rest is Julia set which is nowhere dense (by result of Fatou, it is either the whole plane or nowhere dense).

All this can be found in the paper:

Eremenko and Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst Fourier, 42 (1992) 989-1020.

All these results are classic, (due to Pierre Fatou) except the No-Wandering-Domains theorem which was proved in the paper cited above, and independently by Goldberg and Keen and (for the exponential function) by Baker and Rippon.

Remark. I believe that one can give a completely elementary proof (not using the No Wandering Domains theorem) by the arguments in the paper by Baker, Limit functions and sets of non-normality in iteration theory. Ann. Acad. Sci. Fenn. Ser. A I No. 467 1970 11 pp. But I currently have no access to this paper, so I cannot check. The idea is that when the only singular value is attracted to a fixed point, there can be no wandering domains, and this is elementary. Another, more easily available reference for an elementary proof is the paper of Bergweiler et al. On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 369–375.

Answer 2

Here is one possible elementary argument (somewhat inspired by my paper with Shen in the Monthly, "The exponential map is chaotic"), which avoids any mention of the classification of Fatou components, and indeed of Fatou and Julia sets.

As you mention, the left half-plane is mapped inside itself, and inside the parabolic basin. So you only need to show that the set of points that never enters this half-plane has no interior.

Consider the slit plane $U:=\mathbb{C}\setminus (-\infty,0]$, and its hyperbolic metric. Then the function $f$ strictly expands this metric near every point of $f^{-1}(U)$. Moreover, the expansion factor is bounded away from one away from the interval $(-infty,0]$.

On the other hand, suppose that there was a small disc $D$ all of whose images stayed in the right half-plane. Then it follows that the hyperbolic derivative of $f^n$ as a map from $D$ to $f^n(D)$ is bounded from above; since $f^n(D)\subset U$, the same is true for the hyperbolic derivative as a map $D\to U$. By the above, this means that $f^n(z)$ converges to $0$ as $n\to\infty$, which is impossible by our assumption.