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) = 4x(1-x) 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 = 1, can again be used to define an x satisfying f(x)=4x(1-x). The Lee-Yang zeros no longer lie on the unit circle in the complex z = eh plane for q?2, but their locus still maps onto the Julia set of the logistic map. ©2002 The American Physical Society.