The Fixed Point of a Parametric Discrete Mapping and the Parameter Value Set Ensuring Its Uniqueness

In this paper, we study the set of values of a non-negative integer parameter n for each of which a nonlinear discrete digit-counting mapping Fn:N10→N10, defined by a specific formula Fn(x0,x1,…,x9)=f0(n,x0,…,x9),f1(n,x0,…,x9),…,f9(n,x0,…,x9), where fd(n,x0,…,x9)=1+count(d,n)+count(d,x0)+count(d,x1)+…+count(d,x9), count(d,k) is the number of digits equal to the digit d∈{0,1,…,9} in the decimal notation of the number k∈{0}∪N, possesses a unique fixed point. The fixed point of this mapping is interconnected with a certain self-describing table containing the parameter n. As a result of the study, firstly, an a priori bilateral estimate for the sum of the coordinates of the fixed point is obtained. The bounds of this estimate are determined solely by the parameter n and are achieved simultaneously for certain values of n. Secondly, it is established that if n belongs to a special infinite set described in this work, the last coordinate of the fixed point of this mapping is uniquely determined based on this bilateral estimate. Subsequently, the remaining coordinates of the fixed point are also uniquely found by means of a small exhaustive search. This justifies the infinity of the set of values of n for which the given nonlinear discrete mapping has a unique fixed point, and refutes the hypothesis that the number of fixed points of this mapping increases with n starting from a certain value n0.