Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...ababababab...". The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, "...ababababab...", there are 2 subwords of length 3, namely "aba" and "bab". Since 2 is less than 3, we must have that "...ababababab..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. Interestingly, there are many extensions of this theorem to multiple dimensions and beyond. We prove a few results in two-dimensions, including a specific partial result of a question known as the Nivat conjecture. We also consider a novel extension to the more general setting of 'group actions', and we prove an optimal analogue of the Morse-Hedlund theorem in this setting.
Blaisdell, Eben, "Extensions of the Morse-Hedlund Theorem" (2018). Honors Theses. 463.