We study the convergence to equilibrium states forcertain non-hyperbolic piecewise invertible systems.The multi-dimensional maps we shall considerdo not satisfy Renyi's condition (uniformlybounded distortion for any iterates) and do not necessarilysatisfy the Markov property.The failure of both conditions may cause singularities ofdensities of the invariant measures, even if they arefinite, and causes a crucial difficulty in applying thestandard technique of the Perron–Frobeniusoperator. Typical examples of maps we consideradmit indifferent periodic orbits and arise in many contexts.For the convergence of iterates ofthe Perron–Frobenius operator,we study continuity of the invariant density.