Avoiding and Enforcing Repetitive Structures in Words

Avoiding and enforcing repetitions in words are central topics in the area of combinatorics on words, with first results going back to the beginning of the 20th century.
The results presented in this thesis extend and enrich the existing theory concerning the presence and absence of repetitive structures in words.
In the first part the question whether such structures necessarily appear in infinite words over a finite alphabet is investigated.
In particular, avoidability questions of patterns whose repetitive structure is disguised by the application of a permutation are studied.
The second part deals with equations on words that enforce a certain repetitive structure involving involutions in their solution set.
A generalisation of the classical equations u^l = v^mw^n that were studied by Lyndon and Schützenberger is analysed.
The last part considers the influence of the shuffle operation on square-free words and related avoidability questions.

Mike Müller received his degree in Computer Science at the University of Stuttgart
in 2011.
From 2011 to 2014 he was research assistant in the Dependable Systems Group at the Kiel University, working on the project ``Combinatorial Aspects of Words and their Applications'' funded by the German Science Foundation (DFG).