In this talk, we first review some basic results of square stochastic matrices concerning the convergence of power of square stochastic matrix. We also discuss a linear operator associated with a square stochastic matrix. This gives an advantage to interpret all results by means of the language of dynamical system.
In the second part of this talk, we discuss on a cubic stochastic matrix. Unlike square matrices, there is no proper multiplication of two cubic matrices in which the dimension of matrices would remain invariant. That’s why we can’t speak about the inverse of cubic matrix (which is the important concept) and the power of cubic matrices. However, for each cubic stochastic matrix we can associate a quadratic operator acting on the finite dimensional simplex.
By this way, we can consider the similar problem which was studied in the linear case. For example, due to the Perron-Frobenius theorem, the trajectory of linear operator associated with the positive square stochastic matrix converges to a unique fixed point. It is natural to ask the similar question for quadratic operators, i.e., does the trajectory of quadratic operator associated with the positive cubic stochastic matrix converge to a unique fixed point? However, this is wrong in general. The main reason is that a quadratic operator associated with positive cubic stochastic matrix may have many fixed points. The first attempt to provide an example for quadratic stochastic operators with positive coefficients having three fixed points was done by A.A. Krapivin (Lyubich’s former graduate student) and Y.I. Lyubich. However, it turns out that their examples were wrong.
We showed that Krapivin-Lyubich examples have a unique fixed point. Moreover, we provide an example for a family of quadratic stochastic operators with positive coefficients which have three fixed points. Consequently, the question mentioned above does not have an affirmative answer in general. If it is a case, can we find an affirmative answer for some class of quadratic operators? The last but not least, we also provide a class of quadratic stochastic operators (which are not contraction) in which their trajectory converges a unique fixed point.