We show that for all infinite sequences \(s\in \mathbb {F}_q^\omega \) , two properties are preserved under forward and backward application of the continued fraction operator K (the modified Berlekamp-Massey Algorithm). The first preserved property is that if \({\text {supp}}(s)\subset [r]_n\) , that is, the positions of the nonzero elements of s lie in a certain residue class modulo n, then also \({\text {supp}}(\textbf{K}(s))\subset [r]_n\) . The other property applies only to fields with characteristic two: if the sequence consists of symbol pairs \((s_{2n-1},s_{2n}\) ) with \(s_{2n} = \alpha s_{2n-1}\) for a fixed \(\alpha \in \mathbb {F}_{2^k}\) , for all \(n\in \mathbb {N}\) and \(t:= \textbf{K}(s)\) , then also \(t_{2n} = \alpha t_{2n-1}\) for all \(n\in \mathbb {N}\) . We furthermore determine all sets \(V\subset \mathbb {F}_q^m\) invariant under K for certain finite fields, that is, for which \(\textbf{K}:V^\omega \rightarrow V^\omega \) and conjecture that there are no others even in the general case. In the binary case, \(\mathbb {F}_{2^k}\) , we apply the result to a certain binary tree associated with the isometry K.
