Parametrization for stationary patterns of the r-majority operators on 0-1 sequences

For a nonnegative integer p and a two-way infinite sequence u=(u_m)_m∈Z of complex numbers define the sequence v = A_pu by v_m=(A_pu)_m= ∑ -p≤j≤p(-1)^p+ju_m+j (m∈Z). The spaces U_p and V_p of the sequences u fixed by A²_p and A_p are studied, and given several parametrizations. These spaces play a fundamental role in the description of two-way-infinite 0-1 sequences that reoccur - and, in particular, that are fixed - by the r-majority operators M. M simultaneously replaces every bit of a two-way-infinite 0-1 sequence by the majority bit of the (2r + 1)-segment it centers [2, 9].