Abstract
It is well-known that a linear stochastic operator (a Markov operator) associated with a square stochastic matrix is a surjection of the simplex if and only if it is bijective. The similar problem was open for nonlinear stochastic operators (nonlinear Markov operators) associated with stochastic hyper-matrices (higher dimensional matrices). In this paper, we solved this problem for quadratic stochastic operators acting on the simplex. Namely, we showed that a quadratic stochastic operator associated with a cubic stochastic matrix is a surjection of the simplex if and only if it is bijective. Moreover, we also described all surjective quadratic stochastic operators of the simplex.
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