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Abstract
We study the geometry of representations of numbers by continued fractions whose elements belong to the set A 2 = {α1, α2} (A 2-continued fraction representation). It is shown that, for α1α2 ≤ 1/2 , every point of a certain segment admits an A 2-continued fraction representation. Moreover, for α1α2 = 1/2, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose A 2-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its A 2-continued fraction representation form a homogeneous Markov chain are also investigated.
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