The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period ≤ for a nonlinear transformation

The authors prove that the one-site distribution of Gibbs states (for any finite spin set S) on the Bethe lattice is given by the points satisfying the equation π = T2π, where T = h circ A circ var phi, with var phi(x) = x(q-1)/q, h(x) = (x/parallel x parallel q)q, A = (a(r, s): r, s element-of S), and a(r, s) = exp(K(R, s) + (1/q)(N, r + s)). They also show that for A a symmetric, irreducible operator the nonlinear evolution on probability vectors x(n + 1) = Ax(n)p/parallel Ax(n)p parallel 1 with p > 0 has limit points ξ of period ≤ 2. They show that A positive definite implies limit points are fixed points that satisfy the equation Aξ0 = λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques