Flexible toggles and symmetric invertible asynchronous elementary cellular automata

Our main result is the complete classification of the dynamics of symmetric invertible SDS defined over cycle graphs using the set of states F2 and the identity update order π=123⋯n. More precisely, if T denotes the SDS map of such an SDS, then we obtain an explicit formula for |Perr(T)|, the number of periodic points of T of period r, for every positive integer r. It turns out that if we fix r and vary n and T, then |Perr(T)| only takes at most three nonzero values.