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A divergent horocycle in the horofunction compactification of the Teichmüller metric

We give an example of a horocycle in the Teichmuller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmuller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichmuller space of complex dimension greater than one via covering constructions.