Automatic sequences can be equivalently defined in terms of finite automata (they are produced by them), substitutions (they are their fixed points), or by their kernel (which is finite). The book studies a generalized concept of automatic sequences, where the sequence is indexed by elements of a finitely generated group, instead of natural numbers. Accordingly, ordinary substitutions are replaced by mappings based on expanding group endomorphisms, and a corresponding notion analogous to finite automata is developed. These concepts are introduced and discussed in the first three chapters. Chapter 4 studies automatic subsets and automatic maps. The final chapter combines automatic sequences with some additional algebraic structures, and deals especially with Mahler equations over the ring of formal power series. The book is written quite succinctly, and is not very easy to read. A certain degree of mathematical maturity is required just to grasp the basic definition of the automatic sequence. The volume will be useful mainly for advanced researchers in the field, interested in broadening their view of the subject.

Reviewer:

šh