On periodic properties of circular words

The conjugacy relation defines a partition of words into equivalence classes. We call these classes circular words. Periodic properties of circular words are investigated in this article. The Periodicity Theorem of Fine and Wilf does not hold for weak periods of circular words; instead we give a strict upper bound on the length of a non-unary circular word that has two given relatively prime weak periods. Weak periods also lead to a way of representing circular words in a more compact form. We investigate in which cases are these representations unique or minimal. We will also analyze weak periods of circular Thue–Morse, Fibonacci and Christoffel words.