Quadratic Operators Defined on a Finite-dimensional Simplex of Idempotent Measures

We describe some quadratic operators which map the (n−1) - dimensional simplex of idempotent measures to itself. Such operators are divided to two classes: the first class contains all n×n×n - cubic matrices with nonpositive entries which in each n×n dimensional k-th matrix contains exactly one non-zero row and exactly one non-zero column; the second class contains all n×n×n - cubicmatrices with non-positive entries which has at least one quadratic zero-matrix. These matrices play a role of the stochastic matrices in the case of idempotent measures. For both classes of quadratic maps we find fixed points.

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