Words with unbounded periodicity complexity
\v{S}t\v{e}p\'an Holub
TL;DR
This paper investigates the behavior of periodicity complexity in infinite words, showing that non-periodic words with certain recurrence properties have unbounded complexity, and some can have arbitrarily high complexity at infinitely many positions.
Contribution
It establishes that non-periodic uniformly recurrent words or those with bounded repetition have infinite periodicity complexity, and constructs examples with arbitrarily high complexity at infinitely many positions.
Findings
Non-periodic uniformly recurrent words have infinite periodicity complexity.
Words with bounded repetition also have infinite periodicity complexity.
Some uniformly recurrent words exhibit arbitrarily high complexity at infinitely many positions.
Abstract
If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at infinitely many positions.
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