An analysis of a family of rational maps containing integrable and non-integrable difference analogue of the logistic equation

Abstract

The iteration of rational maps of degree two is discussed for functions which arise from a difference method approximating the logistic equation and we interpolate between integrable and nonintegrable maps. We obtain a necessary and sufficient condition for the statement that the Julia set is contained in the set of real numbers and a sufficient condition for the Julia set to be a Cantor set. The information dimensions of the Julia sets are calculated numerically to show how they change as the parameter of the interpolation approaches the integrable limit.