Escaping set

History

The iteration of transcendental entire functions was first studied by Pierre Fatou in 1926 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 who used Wiman-Valiron theory. |author-link=Alexandre Eremenko journal=Banach Center Publications, Warsawa, PWN|volume=23|pages=339–345|doi=10.4064/-23-1-339-345 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.{{cite journal|first1=G|last1=Rottenfußer| first2=J|last2=Rückert|first3=L|last3=Rempe|author3-link=Lasse Rempe|first4=D|last4=Schleicher| title=Dynamic rays of bounded-type entire functions|journal=Ann. of Math.|volume=173|year=2011|pages=77–125| arxiv=0704.3213|doi=10.4007/annals.2010.173.1.3}} In 2021, a paper by Martí-Pete, Rempe, and Waterman constructed a counterexample to Eremenko's conjecture.

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.

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.
• The boundary of the escaping set is exactly the Julia set. In particular, the escaping set is never closed.
• For a transcendental entire function, the escaping set always intersects the Julia set. 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.
• The escaping set always has at least one unbounded connected component.
• The escaping set is connected or has infinitely many components.
• The set I(f)\cup {\infty} is connected. title=Boundaries of escaping Fatou components|journal=Proceedings of the American Mathematical Society|year=2011|volume=139|issue=8|pages=2807–2820|doi=10.1090/s0002-9939-2011-10842-6|arxiv=1009.4450}}

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

Polynomials

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) 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 f(z) = z^2 consists precisely of the complement of the closed unit disc:

:I(f) = {z\in\mathbb{C} \colon |z| 1 }.

Transcendental entire functions

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). 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\delta}\text{ set}. It is neither G_{\delta} nor F_{\sigma}. For functions in the exponential class \exp(z)+a, the escaping set is not G_{\delta\sigma}.

A comprehensive survey of escaping sets of transcendental functions in given in the paper of Bergweiler and Rempe

Notes

References

References

1. Sixsmith, D.J.. (2012). "Entire functions for which the escaping set is a spider's web". [[Mathematical Proceedings of the Cambridge Philosophical Society]].
2. Rempe, Lasse. (2020). "Escaping sets are not sigma-compact". [[Proceedings of the American Mathematical Society]].
3. Lipham, D.S.. (2024). "Exponential Iteration and Borel Sets". Computational Methods and Function Theory.
4. (2025). "The escaping set in transcendental dynamics". Jahrbuch über die Fortschritte der Mathematik.