#### Date of Thesis

Spring 2018

#### Thesis Type

Honors Thesis

#### Degree Type

Bachelor of Science

#### Major

Mathematics

#### Second Major

Computer Science

#### First Advisor

Van Cyr

#### Keywords

Morse-Hedlund, word, complexity, symbolic dynamics, group action

#### Abstract

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.

#### 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