Uniqueness of fixed point of a two-dimensional map obtained as a generalization of the renormalization group map associated to the self-avoiding paths on gaskets

Let W (x, y) = a x + b x + f5 x 5 + f6 x 6 +(3 a x)y+ g5 x y+h3 x y +h4 x y +n3 x y + a24 x y + a05 y 5 + a15 xy 5 + a06 y , and X = ∂W ∂x , Y = ∂W ∂y , where the coefficients are non-negative constants, with a > 0, such that X(x, x) − Y (x, x) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ : (x, y) 7→ (X(x, y), Y (x, y)) satisfying the conditions are the renormalization group (RG) map (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point (xf , yf) of Φ in the invariant set {(x, y) ∈ R+ 2 | x ≧ y} \ {0}.