Multidimensional Sturmian sequences and generalized substitutions

This paper is devoted to a study on the way generalized substitutions - a multidimensional extension of substitutions - act on multi-dimensional Sturmian sequences. We give a sufficient condition under which these multi-dimensional Sturmian sequences are obtained by iterated compositions of generalized substitutions. This condition relies on Brun expansions - a multi-dimensional extension of continued fraction expansions.