Subword occurrences, weighted automata and iterated morphisms, especially the Fibonacci morphism

The paper investigates the number of subword occurrences in a word, in particular, in a word obtained by iterating a morphism. Associated graphs or automata are used as a tool. Special attention is paid to the Fibonacci morphism. Explicit formulas in terms of n are obtained for the number of occurrences of subwords of length at most 2 in the nth iteration of the Fibonacci morphism.