Geodesic
Determination of Periods of Geometric Continued Fractions for Two-Dimensional Algebraic Hyperbolic Operators
Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 30-42

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An explicit construction of a reduced hyperbolic integer operator from the group $SL(2,\mathbb Z)$ such that one of the periods of the corresponding geometric continued fraction in the sense of Klein coincides with a given sequence of positive integers is presented. An algorithm determining periods for any operator in $SL(2,\mathbb Z)$ (which is based on Gauss' reduction theory) is experimentally studied.
Keywords: geometric continued fraction in the sense of Klein, period of a geometric continued fraction, hyperbolic integer operator, sail of an integer operator, LLS-sequence, integer length, integer sine. 
@article{MZM_2010_88_1_a2,
     author = {O. N. Karpenkov},
     title = {Determination of {Periods} of {Geometric} {Continued} {Fractions} for {Two-Dimensional} {Algebraic} {Hyperbolic} {Operators}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {30--42},
     publisher = {mathdoc},
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     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a2/}
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O. N. Karpenkov. Determination of Periods of Geometric Continued Fractions for Two-Dimensional Algebraic Hyperbolic Operators. Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 30-42. http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a2/