z0\inC
zn+1:=f(zn)
n
f
I(f)
For example, for
f(z)=ez
0,1,e,ee,e
| ee | |
,...
The iteration of transcendental entire functions was first studied by Pierre Fatou in 1926[2] The escaping set occurs implicitly in his study of the explicit entire functions
f(z)=z+1+\exp(-z)
f(z)=c\sin(z)
The first study of the escaping set for a general transcendental entire function is due to Alexandre Eremenko who used Wiman-Valiron theory.[3] He conjectured that every connected component of the escaping set of a transcendental entire function is unbounded. This has become knownas Eremenko's conjecture.[4] There are many partial resultson this problem but as of 2013 the conjecture is still open.
Eremenko also asked whether every escaping point can be connected to infinity by a curve in the escaping set; it was later shown that this is not the case. Indeed,there exist entire functions whose escaping sets do not contain any curves at all.
The following properties are known to hold for the escaping set of any non-constant and non-linear entire function. (Here nonlinear means that the function is not of the form
f(z)=az+b
f
I(f)\cup\{infty\}
Note that the final statement does not imply Eremenko's Conjecture. (Indeed, there exist connected spaces in which the removal of a single dispersion point leaves the remaining space totally disconnected.)
A polynomial of degree 2 extends to an analytic self-map of the Riemann sphere, having a super-attracting fixed point at infinity. The escaping set is precisely the basin of attraction of this fixed point, and hence usually referred to as the **basin of infinity**. In this case,
I(f)
f(z)=z2
I(f)=\{z\inC\colon|z|>1\}.
For transcendental entire functions, the escaping set is much more complicated than for polynomials: in the simplest cases like the one illustrated in the picture it consists of uncountably many curves, called hairs or rays. In other examples the structure of the escaping set can be very different (a spider's web).[6] As mentioned above, there are examples of transcendental entire functions whose escaping set contains no curves.
By definition, the escaping set is an F\sigma\deltaset G\delta F\sigma
. It is neither
nor
.[7] For functions in the exponential class
\exp(z)+a
G\delta\sigma