Geodesic
The spectral type of the rearrangements $T_{\alpha,\beta}$
Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1119-1152

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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. 
@article{SM_1997_188_8_a1,
     author = {O. N. Ageev},
     title = {The spectral type of the rearrangements $T_{\alpha,\beta}$},
     journal = {Sbornik. Mathematics},
     pages = {1119--1152},
     publisher = {mathdoc},
     volume = {188},
     number = {8},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_8_a1/}
}
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O. N. Ageev. The spectral type of the rearrangements $T_{\alpha,\beta}$. Sbornik. Mathematics, Tome 188 (1997) no. 8, pp. 1119-1152. http://geodesic.mathdoc.fr/item/SM_1997_188_8_a1/