Geodesic
Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers
Matematičeskie trudy, Tome 10 (2007) no. 1, pp. 97-131

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The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated. 
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V. N. Berestovskii; Yu. G. Nikonorov. Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers. Matematičeskie trudy, Tome 10 (2007) no. 1, pp. 97-131. http://geodesic.mathdoc.fr/item/MT_2007_10_1_a3/