" The structure of the fixed point set of quadratic operators on the simplex"

We know that any linear operator associated with a positive square stochastic matrix has a unique fixed point in the simplex. However, in general, the similar result for a quadratic operator acting on the simplex does not hold true. Namely, there is a quadratic operator associated with a positive cubic stochastic matrix which has more than one fixed point in the simplex. The first attempt to give an example for such kind of quadratic operators was done by A.A. Krapivin and Yu.I. Lyubich. However, we showed that their examples are wrong. Therefore, in this paper, we decided to give a correct example for a quadratic operator with positive coefficients having three fixed points in the simplex. Moreover, we also describe the number of fixed points of the quadratic operator associated with a positive cubic stochastic matrix.

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