On the complete chaotic transformations that preserve symmetric invariant densities

A transformation f:[0,1]→[0,1] is said to be complete chaotic if it is (i) ergodic with respect to the Lebesgue measure and (ii) chaotic in the probabilistic sense, that is, an absolutely continuous invariant density φ is preserved. The characteristics of the complete chaotic transformations that preserve symmetric invariant densities, that is, φ(x)=φ(1-x), for all x [0,1], are explored. It is found that such transformations are “invariant” with both horizontal and vertical mirroring operations in the sense that the transformations resulted do not only remain to be chaotic but also preserve an identical invariant density. Numerical examples and computer simulations are consistent with theoretical findings.