In this paper, we study a class of cellular automata (CA) called stable cellular automata (SCA) that preserve stability by reflection, modulo-recurrent, and richness. After applying these automata to Sturmian words, we determine some of their combinatorial properties. Next, we calculate the classical and palindromic complexity functions of these words. Finally, we demonstrate that these words are $2$-balanced and establish their abelian complexity function.

17 pages, 0 figure, to be published in Discrete Mathematics and Theoretical Computer Science (DMTCS)