Distribution of Values of Cantor Type Fractal Functions with Specified Restrictions

We consider \(Q_s^*\)-representation of numbers \(x \in [0,1]\). It is an encoding of real numbers by means of the finite alphabet \(A=\{0,1,2,\ldots ,s-1\}\). The article is devoted to continuous non-monotonic singular functions of Cantor type defined in terms of a given \(Q_s^*\)-representation of numbers. We study their local and global properties: structural, variational, differential, integral, self-similar, and fractal. Level sets of functions as well as topological and metric properties of images of Cantor type sets are examined in detail. In this work we also study the distribution of random variable \(Y=f(X)\), where f is a non-monotonic singular function of Cantor type and X is a random variable such that its distribution induced by distributions of digits of its \(Q_5^*\)-representation that are independent random variables.