Fully developed chaotic 1−d maps

Single-hump 1–d maps are investigated which generate ergodic process on an interval mapped everywhere two-to-one onto itself. Introducing a new transformation transverse to conjugation it is shown that such maps are related by smooth transformations to each other. It is found that each of the families consisting of conjugate maps contains a map everywhere expanding and producing ergodic iterations according to the uniform probability density. The general framework is used to construct maps together with their probability density functions. Quantities characterizing the dynamics are calculated and their parameter dependence while maintaining the fully developed chaotic state is studied. Furthermore, universal maps exhibiting fully developed chaos are considered.