The spectral type of the rearrangements $T_{\alpha,\beta}$

A method of geometric models is proposed and applied to the study of the spectral properties of the classical transformations $T_{\alpha,\beta}$. It is proved that the class of ergodic transformations under consideration with absolutely continuous and mixing components contains no transformation with a non-simple spectrum. A criterion for the ergodicity of the transformations $T_{\alpha,\beta}$ is obtained in terms of the geometric models. The multiplicity function of the spectrum of $T_{\alpha ,\beta}$ is determined for any $n$ when $\alpha$ is the golden section.