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A new classification scheme for pairs of real numbers is given, generalizingearlier work of the author that used continued fraction, which in turn wasmotivated by ideas from statistical mechanics in general and work of Knauf andFiala and Kleban in particular. Critical for this classification are the numbertheoretic and geometric properties of the triangle map, a type ofmulti-dimensional continued fraction.

A thermodynamic classification of pairs of real numbers via the Triangle Multi-dimensional continued fraction