Effects of randomization on asymptotic periodicity of nonsingular transformations

It is known that the Perron–Frobenius operators of piecewise expanding ${\mathcal{C}}^2$ transformations possess an asymptotic periodicity of densities. On the other hand, external noise or measurement errors are unavoidable in practical systems; therefore, all realistic mathematical models should be regarded as random iterations of transformations. This paper aims to discuss the effects of randomization on the asymptotic periodicity of densities.