Inspired by the celebrated Lothaire series (Lothaire, 1983, 2002, 2005) and animated by the same spirit as in the book (Berthé and Rigo, 2010), this collaborative volume aims at presenting and developing recent trends in combinatorics with applications in the study of words and in symbolic dynamics.
On the one hand, some of the newest results in these areas have been selected for this volume and here benefit from a synthetic exposition. On the other hand, emphasis on the connections existing between the main topics of the book is sought. These connections arise, for instance, from numeration systems that can be associated with algorithms or dynamical systems and their corresponding expansions, from cellular automata and the computation or the realisation of a given entropy, or even from the study of friezes or from the analysis of algorithms.
This book is primarily intended for graduate students or research mathematicians and computer scientists interested in combinatorics on words, pattern avoidance, graph theory, quivers and frieze patterns, automata theory and synchronised words, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory and transfer operators, numeration systems, dynamical arithmetics, analytic combinatorics, continued fractions, probabilistic models. We hope that some of the chapters can serve as usefulmaterial for lecturing at master/graduate level. Some chapters of the book can also be interesting to biologists and researchers interested in text algorithms or bio-informatics.
Let us succinctly sketch the general landscape of the volume. Short abstracts of each chapter can be found below. The book can roughly be divided into four general blocks. The first one, made of Chapters 2 and 3, is devoted to numeration systems. The second block, made of Chapters 4 to 6, pertains to combinatorics of words. The third block is concernedwith symbolic dynamics: in the one-dimensional setting with Chapter 7, and in the multidimensional one, with Chapters 8 and 9. The last block, made of Chapters 10 and 11, has again a combinatorial nature.
Words, i.e., finite or infinite sequences of symbols taking values in a finite set, are ubiquitous in the sciences. It is because of their strong representation power: they arise as a natural way to code elements of an infinite set using finitely many symbols. So let us start our general description with combinatorics on words.