The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin ${\sigma }_i\in \pm 1$. The $q$-state Potts model is a generalization of the Ising model where each spin ${\sigma }_i$ may take on $q\ge 3$ number of states $0,\cdots ,q-1$. Both models have temperature $T$ and an externally applied magnetic field $h$ as parameters. Many statistical and physical properties of the $q$-state Potts model can be derived by studying its partition function. This includes phase transitions as $T$ and/or $h$ are varied.

The celebrated Lee-Yang Theorem characterizes such phase transitions of the $2$-state Potts model (the Ising model). This theorem does not hold for $q>2$. Thus, phase transitions for the Potts model as $h$ is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the $3$-state Potts model as $h$ is varied for constant $T$ on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed $T>0$ the $3$-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of $h$ or not at all, depending on $T$. However, an interesting new phenomenon occurs for the $3$-state Potts model because the critical value of $h$ can be non-zero for some range of temperatures. The $3$-state Potts model for the antiferromagnetic case exhibits phase transition at up to two critical values of $h$.

The recursive constructions of the $(n+1{)}^{st}$ level Cayley tree from two copies of the $n^{th}$ level Cayley tree allow one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.