On fixed points of strictly positive stochastic operators on a one-dimensional simplex

A class of strictly positive stochastic operators of degree n acting on a one-dimensional simplex is investigated. The main focus is on the problem of the number of fixed points of such operators. It is shown that this problem reduces to studying the number of roots of the corresponding degree n polynomial on the interval (0 , 1). The existence of at least one fixed point is proved for arbitrary n . For the quadratic case ( n = 2), the uniqueness of the fixed point is established. For a cubic operator ( n = 3), conditions for the existence of one, two, or three fixed points are obtained. For a quartic operator ( n = 4), criteria determining the number of fixed points are proved in terms of the operator’s parameters.

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