Density function of piecewise linear transformation

Let fa,b(x)=minax+a+b−ab/b,b(1−x): I → I be a piecewise linear map from the unit interval, I = [0, 1] into itself where a > 0 and b > 1. This map can generate various dynamics ranging from the simple to complex involving chaos according to the strength of nonlinearities involved. This study clarifies parameter combinations for which the map can generate periodic cycles and then constructs explicit forms of density functions that are useful to derive various statistical properties of chaotic trajectories. Two illustrative examples are given to show the effectiveness of our results.