Se p 20 07 The Number of Periodic Orbits of a Rational Difference Equation

where g is a rational function of two variables, which we write as a quotient of polynomials without common factor. Given starting values z0 = x and z1 = y, equation (0.1) gives rise to an infinite sequence (zn)n≥0, as long as the denominator in (0.1) does not vanish. One of the basic questions here is to find the periodic sequences generated by (0.1), i.e., sequences for which zj+p = zj for all j. We refer to the books [KL], [KM] and [GL] for expanded discussions of this question. Here we describe a method for giving an upper bound on the complex periodic sequences of period p. We note that equations (0.1) have been widely studied in the real domain, frequently for zn > 0; when we obtain an upper bound on the number of complex periodic points, this upper bound also applies to the periodic sequences which are real (or positive). Let us write (0.1) as a rational map