Escaping set

In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ.[1] That is, a complex number $z_0\in\mathbb{C}$ belongs to the escaping set if and only if the sequence defined by $z_{n+1} := f(z_n)$ converges to infinity as $n$ gets large. The escaping set of $f$ is denoted by $I(f)$.[1]

For example, for $f(z)=e^z$, the origin belongs to the escaping set, since the sequence

$0,1,e,e^e,e^{e^e},\dots$

tends to infinity.

The basin of infinity for polynomials

For a polynomial of degree at least 2, the point at infinity is an (super-)attracting fixed point, and the escaping set is precisely the basin of attraction of this fixed point. Hence in this case, $I(f)$ is an open and connected subset of the complex plane, and the Julia set is the boundary of this basin.

For instance, the escaping set of the complex quadratic polynomial ƒ(z) = z2 consists precisely of those points whose absolute value is greater than 1.

History

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 $f(z)=z+1+\exp(-z)$ and $f(z)=c\sin(z)$.

The first study of the escaping set for a general transcendental entire function is due to Alexandre Eremenko.[3] He conjectured that every connected component of the escaping set of a transcendental entire function is unbounded. This has become known as Eremenko's Conjecture.[1][4] There are many partial results on 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.[4]

Properties

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 $f(z)=az+b$.)

• The escaping set contains at least one point.[3]
• The boundary of the escaping set is exactly the Julia set.[3] In particular, the escaping set is never closed.
• For a transcendental entire function, the escaping set always intersects the Julia set.[3] In particular, the escaping set is open if and only if $f$ is a polynomial.
• Every connected component of the closure of the escaping set is unbounded.[3]
• The escaping set always has at least one connected component.[1]
• The set $I(f)\cup \{\infty\}$ is connected.[5]

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.)

Examples

Escaping set of (exp x − 1)/2. (Picture by Arnaud Cheritat)

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 on 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 entire functions whose escaping set contains no curves.[4]

References

1. ^ a b c d Rippon, P. J.; Stallard, G (2005). "On questions of Fatou and Eremenko". Proc. Amer. math. Soc. 133: 1119–1126.
2. ^ Fatou, P. (1926). "Sur l'itération des fonctions transcendantes Entières". Acta Math. 47: 337–370. doi:10.1007/bf02559517.
3. Eremenko, A (1987). "On the iteration of entire functions". Banach center publications, Warsawa, PWN 23: 339–345.
4. ^ a b c Rottenfußer, G; Rückert, J; Rempe, L; Schleicher, D (2011). "Dynamic rays of bounded-type entire functions". Ann. of Math. 173: 77–125. doi:10.4007/annals.2010.173.1.3.
5. ^ Rippon, P. J.; Stallard, G (2011). "Boundaries of escaping Fatou components". Proc. Amer. math. Soc. 139: 2807–2820. doi:10.1090/s0002-9939-2011-10842-6.
6. ^ Sixsmith, D.J. (2012). Entire functions for which the escaping set is a spider's web.