Dynamical systems and ergodic theory Complex dynamics Conjecture

Eremenko's conjecture

Builds on Entire function Julia set Connected component

Statement

Let $f \colon \C \to \C$ be a transcendental entire function. Let $I(f) = \set{z \in \C : f^n(z) \to \infty \text{ as } n \to \infty}$ be its escaping set. Then every connected component of $I(f)$ is unbounded.

Remarks

Eremenko proved that $I(f)$ is nonempty and that the Julia set of $f$ is the boundary of $I(f)$ . He also proved that every component of the closure of $I(f)$ is unbounded. Rippon and Stallard proved in 2005 that $I(f)$ always has at least one unbounded component. A stronger form, that every escaping point can be joined to $\infty$ by a curve in $I(f)$ , is false. Rottenfußer, Rückert, Rempe and Schleicher gave entire functions whose escaping sets contain no curves at all. In 2021 Martí-Pete, Rempe and Waterman constructed a counterexample to the conjecture itself. For a polynomial of degree at least $2$ the analogous statement holds. There the escaping set is the complement of the filled Julia set $K$ , which is compact and has connected complement. So the escaping set is connected and unbounded.

Reading path

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Tag	`WWSYQJF7TZ`
URI	`m2://WWSYQJF7TZ`
Kind	conjecture
Writing	W0 stub
MSC	37F10, 30D05
Status	This copy of the site has no build data. Run `m2 status WWSYQJF7TZ`.
About	Entire function
Uses	Entire function, Julia set, Connected component

Eremenko's conjecture

Statement

Remarks

Reading path

Further reading

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