Decomposition theorem for invertible substitutions on three-letter alphabet

Abstract

We shall characterize the structure of invertible substitutions on three-letter alphabet. We show that any invertible substitution, after some cyclic operation, can be written as a finite product of permutations and Fibonacci's substitution. As a consequence, a matrix (of order 3 and with non-negative integral coefficients) is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.