As the title may suggest, this book is about combinatorics on words, automata and formal language theory, as well as number theory. This collaborative work gives a glimpse of the active community working in these interconnected and even intertwined areas. It presents several important tools and concepts usually encountered in the literature and it reveals some of the exciting and non-trivial relationships existing between the considered fields of research. This book is mainly intended for graduate students or research mathematicians and computer scientists interested in combinatorics on words, theory of computation, number theory, dynamical systems, ergodic theory, fractals, tilings and stringology. We hope that some of the chapters can serve as useful material for lecturing at master level.

The outline of this project has germinated after a very successful international eponymous school organised at the University of Liége (Belgium) in 2006 and supported by the European Union with the help of the European Mathematical Society (EMS). Parts of a preliminary version of this book were used as lecture notes for the second edition of the school organized in June 2009 and mainly supported by the European Science Foundation (ESF) through the AutoMathA programme. For both events, we acknowledge also financial support from the University of Liége and the Belgian funds for scientific research (FNRS).

We have selected ten topics which are directed towards the fundamental three themes of this project (namely, combinatorics, automata and number theory) and they naturally extend to dynamical systems and ergodic theory (see Chapters 6 and 7), but also to fractals and tilings (see Chapter 5) and spectral properties of matrices (see Chapter 11).