Hierarchy of piecewise non-linear maps with non-ergodic behaviour

We study in detail the dynamics of a hierarchy of piecewise maps generated by a one-parameter family of trigonometric chaotic maps and a one-parameter family of elliptic chaotic maps of the cn and sn type. We calculate the Lyapunov exponent and Kolmogorov-Sinai entropy of the these maps with respect to a control parameter. Non-ergodicity of these piecewise maps is proven analytically and investigated numerically. The invariant measure of these maps, which are not equal to 1 or 0, appears to be characteristic for non-ergodic behaviour. A quantity of interest is the Kolmogorov-Sinai entropy, which, for these maps, is smaller compared with the sum of the positive Lyapunov exponents, and it confirms the non-ergodicity of the maps