Periodic Points of Quadratic Polynomials

A point of period k of a function f is a point x such that f (x) = x but f (x) 6= x for any 0 < i < k. Let φk(f) denote the number of real points of period k of f . In the case that f is a quadratic polynomial, we develop an algorithm to explicitly characterize φk(f) in terms of the coefficients of f. We show the resulting characterization for k ≤ 6. The method extends to higher k and higher degrees of f but becomes computationally intensive.