Hierarchy of piecewise non-linear maps with non-ergodicity behavior
  We study the dynamics of hierarchy of piecewise maps generated by one-parameter families of trigonometric chaotic maps and one-parameter families of elliptic chaotic maps of $\mathbf{cn}$ and $\mathbf{sn}$ types, in detail. We calculate the Lyapunov exponent and Kolmogorov-Sinai entropy of the these maps with respect to 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 one or zero, appears to be characteristic of non-ergodicity behavior. A quantity of interest is the Kolmogorov-Sinai entropy, where for these maps are smaller than the sum of positive Lyapunov exponents and it confirms the non-ergodicity of the maps. 
M.A. Jafarizadeh, M. Foroutan and S. Behnia