Quadratic stochastic operators, Lyapunov functions, and tournaments

A class of quadratic stochastic operators acting in a finite-dimensional simplex is distinguished that has trajectory at any point of the simplex behaving in a nonregular fashion as a rule. For the discrete dynamical systems generated by such operators the existence of a Lyapunov function of the form $\varphi=x_1^{p_1}\dots x_m^{p_m}$ is proved, and an algorithm for finding the numbers $p_1,\,\dots,\,p_m$ is indicated. Upper estimates are obtained for the set $\omega(x^0)$ of limit points of the trajectories. It is proved that the 'negative' trajectories exist and converge. The question of the number of isolated fixed points of the operators in the distinguished class is considered. The connection between discrete dynamical systems and the theory of tournaments is also studied.