Julia sets associated with the Potts model on the Bethe lattice and other recursively solved systems
Abstract
The q-state Potts model on the Bethe lattice has been investigated by a number of individuals over the last several decades with some of the preceding studies taking a dynamical systems perspective. However, other than for the special case of the q = 2 Potts model, i.e. the Ising model, the Julia sets of the discrete dynamical systems associated with the Potts model on the Bethe lattice have been largely ignored. We look at these sets and find an interesting connection between the phase transition temperature and the dimension of the Julia set. In particular we find the dimension of the Julia set to be a minimum at a phase transition. Furthermore we show this property is not restricted to Potts models on Bethe lattices. This adds to a number of other special properties present at temperatures where a phase transition occurs.