Potts model on a Cayley tree and logistic equation

The q-state Ports model on a Cayley tree can be solved by a recursion formula depending on the properties at the surface. The model on a Bethe lattice is obtained by extrapolating the interior of a Cayley tree sufficiently far from the surface in order to have a stable fixed point of the recursion. For q > 2 we find a second-order transition of percolation type at the Bethe-Peierls temperature and a first-order transition at a higher temperature. For coordination number z = 3 the recursion extrapolated to q = 1 is identical to the logistic equation. The Feigenbaum route to chaos appears for antiferromagnetic coupling of the Ports model. The first period doubling corresponds to a multicritical point in the phase diagram of the Ports model.