Lasse Rempe-Gillen


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A PROLOGUE IN HEXAMETERS:

Sing us, oh Muse, of the man who first considered those orbits -
under a map of the plane that is transcendental in nature -
that leave all bounded domains in a journey to infinite shores.

Using the swords that were forged by Goerges Valiron and Wiman,
he was able to show that such points will always be present, and
cannot be easily blocked from the view of their ultimate goal.

Daring to go even further, he also raised beautiful questions:
Questions that still to this day we have not been able to answer.
Eremenko is his name, master of points that escape.

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A SOLILOQUY:

Class $\B$ or not class $\B$, that is the question -
Whether 'tis better in the end to suffer
the tracts that lie over unbounded values,
or to take arms against these spots of trouble,
and by assumption ban them?

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A SONNET

Let $f$ a transcendental function be
whose sing'lar points - which we assume are bounded,
as Eremenko-Lyubich once demanded -
immediately to a fixed point flee.

Within its Julia set we then may see
maximal shapes connected and unbounded
which - not being arcs - leave us confounded:
What might their topolog'cal structure be?

My friends, mayhap we ought to light some candles
for those who hoped that only simple sets escape.
Continua in Julia sets that may arise
we can describe and find that they comprise
not only pseudo-arcs and bucket handles,
but any other kind of arc-like shape.