The Boolean averaging procedure, common to three systems in statistical mechanics, control and signal processing

We investigate the Boolean averaging procedure (BAP) u(n) = sgn(ψ Σ_i=1 ^∞ γ_iu(n-i)), where ψ∈R (γ_i) is a weight function (i ∈ N), (u(n)) is a two-sided sequence (n ∈ Z). It describes three systems: 1) the Hubbard model in statistical mechanics, 2) sampled-data control system, and 3) analog-to-digital converter with sigma-delta modulation and leaky integration. The periodic modes in the BAP generate so-called Hubbard configurations, in which (+1)-s is evenly distributed between (-1)-s (the phenomenon of even 2-colouring). Our new results shows that the trajectories (u(n)) become periodic for almost all ψ if the sequence (γ_n) is convex.