Preserving measure chaotic maps of domains in the form of rotation figures

The aim of the article is to demonstrate an algorithm for constructing measure-preserving three-dimensional chaotic maps defined in domains formed by rotation bodies. On the one hand, the class of multidimensional chaotic mappings is expended, and on the other hand, we obtain formulas for simulating pseudo-random quantities that are in demand in problems solving by the Monte Carlo method. The analytical algorithm for constructing multidimensional maps consists of the following steps: 1) the presentation of the invariant density as the product of the unconditional distribution of a point coordinates of the map’s orbit and the conditional densities of the distribution of other coordinates (provided that the values of some coordinates take a fixed value); 2) finding the corresponding integral distribution laws for the coordinates of the point of the mapping; 3) presentation of the coordinates of the point of the orbit through pseudo-random variables by using the inverse function modeling method; 4) reduction of the obtained dependences to the form of chaotic mappings for a particular choice of a chaotic onedimensional map possessing a uniform invariant distribution. The last step allows us to present pseudorandom values as iterative deterministic procedures defined on areas of complex shape. Statistical properties correlate with an array of generated numbers that have the sense of the coordinates of a pseudo-random point in a space bounded by a rotation figure. Examples of the synthesis of three-dimensional chaotic mappings (generators of pseudo-random points) are considered both for the general case (defining the generator of the body of revolution by an arbitrary continuous function) and for specific types of three-dimensional regions in the form of a sphere and a cone. Methods are discussed that allow one to smooth out the property of rationality set of computer numbers by modeling pseudo-random variables.