Date of Thesis
Spring 2018
Description
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.
Keywords
Morse-Hedlund, word, complexity, symbolic dynamics, group action
Access Type
Honors Thesis
Degree Type
Bachelor of Science
Major
Mathematics
Second Major
Computer Science
First Advisor
Van Cyr
Recommended Citation
Blaisdell, Eben, "Extensions of the Morse-Hedlund Theorem" (2018). Honors Theses. 463.
https://digitalcommons.bucknell.edu/honors_theses/463
Included in
Algebra Commons, Discrete Mathematics and Combinatorics Commons, Dynamical Systems Commons, Geometry and Topology Commons, Theory and Algorithms Commons