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 (x)54x(12x) 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 q52, can again be used to define an x satisfying f (x)54x(12x). The Lee-Yang zeros no longer lie on the unit circle in the complex z5eh plane for qÞ2, but their locus still maps onto the Julia set of the logistic map.