It is known that the exact renormalization transformations for the one-dimensional Ising model in a field can be cast in the form of the logistic map $f\left(x\right)=4x\left(1\mathrm{}-x\right)$ with x a function of the Ising couplings K and h. The locus of the Lee-Yang zeros for the one-dimensional Ising model in the $K,h$ plane is given by the Julia set of the logistic map. In this paper we show that the one-dimensional q-state Potts model for $q>~1$ also displays such behavior. A suitable combination of couplings, which reduces to the Ising case for $q=2,$ can again be used to define an x satisfying $f\left(x\right)=4x\left(1\mathrm{}-x\right).\mathrm{}$ The Lee-Yang zeros no longer lie on the unit circle in the complex ${z=e}^h$ plane for $q\ne 2,$ but their locus still maps onto the Julia set of the logistic map.

• Received 22 February 2002

DOI:https://doi.org/10.1103/PhysRevE.65.057103