The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period≤2 for a nonlinear transformation

We 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·A·φ{symbol}, with φ{symbol}(x)=x^(q-1/q, h(x)=(x∥x∥_q)^q, A=(a(r, s):r, s∈S), and {Mathematical expression} We also show that for A a symmetric, irreducible operator the nonlinear evolution on probability vectors x(n+1)=Ax(n)^p∥Ax(n)^p∥₁ with p>0 has limit points ξ of period≤2. We show that A positive definite implies limit points are fixed points that satisfy the equation Aξ^p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.