Invariant functions of an iterative process for maximization of a polynomial

Let P be a polynomial with real non-negative coefficients and variables Xij, i = 1, , k, j = 1, , n. Let d== ^f%. Let Ed be the ^-dimensional real vector space. Let M be the subset of Rd defined by 0, J where the symbols Xij denote the components of x. If $ is a vector in the interior of M, define (x) as the vector in M with components Xi tJ given by dP The expression on the right is evaluated at x. The transformation is defined on the boundary of M by the same formula if the denominators do not vanish. Let F be the set of fixed points of in M. It is shown that if is a homeomorphism of M onto itself, there is a set ofd -k functions /i, ,/<*-& defined on M -F such that /<() = /<W)) f or xeM-F. The functions/< are continuous and independent on an open dense subset of M -F. Explicit expressions for certain invariant functions are also obtained.