Periodic points of positive linear operators and Perron-Frobenius operators

Let C(S) denote the Banach space of continuous, real-valued maps f : S → IR and let A denote a positive linear map of C(S) into itself. We give necessary conditions that the operator A have a strictly positive periodic point of minimal period m. Under mild compactness conditions on the operator A, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal period m. We study a class of Perron-Frobenius operators defined by (Ax)(t) = ∑^∞_i=1 bi(t)x(wi(t)), and we show how to verify the necessary compactness conditions to apply our theorems concerning existence of positive periodic points.