We study the geometry of representations of numbers by continued fractions whose elements belong to the set A 2. It is shown that, for α α 1 2 1 2 ≤ / , every point of a certain segment admits an A 2 -continued fraction representation.Moreover, for α α 1 2 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.

We consider one generalization of functions, which are called as «binary self-similar functi- ons» by Bl. Sendov. In this paper, we analyze the connections of the object of study with well known classes of fractal functions, with the geometry of numerical series, with distributions of random variables with independent random digits of the two-symbol $Q_2$-representation, with theory of fractals. Structural, variational, integral, differential and fractal properties are studied for the functions of this class.