This habilitation thesis is a collection of articles covering various topics,
published or submitted from 2013 to 2016. Most of the articles are set in
Combinatorics on Words and are dealing with languages generated by mor phisms that have reversal symmetry: with each element of the language, the
mirror image of this element is also in the language.
The rst part of the articles investigates some of the recent conjectures
in Combinatorics on Words. The rst conjecture is the Brlek Reutenaurer
conjecture, connecting palindromic defect with factor and palindromic com plexities. We solve this conjecture by giving an a rmative answer. The
second conjecture is the Class P conjecture, stating that if a language gener ated by a morphism contains in nitely many palindromes, then the morphism
belongs to a special class of morphism called class P. The third conjecture is
the Zero defect conjecture, which states that if the generating morphism is
primitive, then the palindromic defect of the language is zero or in nity. We
give only partial answers to the last two conjectures: we deal with some spe ci c subclasses of the morphisms in question. Namely, we give an a rmative
answer for morphisms xing 3 interval exchange transformation for Class P
conjecture, and for binary and primitive marked morphisms for Zero defect
conjecture.
The second part of the articles present many new constructions of words
with nite palindromic defect, also in a generalized sense. We enlarge the
family of known examples of such words by following the construction of
Rote words, by doing letter-to-letter projections of episturmian words, and
by investigating generalized Thue Morse words.
The third part of the thesis deals with e cient algorithmic analysis of
languages generated by morphisms and leads toward two e cient algorithms:
the rst algorithm enumerates all primitive factors that occur in the generated
language in any power; the second algorithm tests whether a morphism is
circular.
The last part is constituted from two various results: the study of the
Rauzy gasket, a set representing letter frequencies of all ternary episturmian
words; and the study of a generalization of Markov constant motivated by the
study of spectrum of a a di erential operator. This part serves as an illustra tion of connection of Combinatorics on Words to other research domains.
