The fixed point of a generalization of the renormalization group maps for self-avoiding paths on gaskets

Let W(x,y) = a x³ + b x⁴ + f₅ x ⁵ + f₆ x⁶ + (3 a x²)² y + g₅ x⁵ y + h₃ x³ y² + h₄ x⁴ y² + n₃ x³ y ³ + a₂₄ x² y⁴ + a₀₅ y⁵ + a₁₅ xy⁵ + a₀₆ y⁶, and X = ∂W/∂x, Y = ∂W/∂y, where the coefficients are non-negative constants, with a > 0, such that X²(x, x²) - Y(x, x²) is a polynomial of x with non-negative coefficients. Examples of the 2 dimensional map Φ: (x, y) → (X(x, y), Y(x, y)) satisfying the conditions are the renormalization group (RG) maps (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 (x_f, y_f) of Φ in the invariant set {(x, y) ∈ ℝ ₊² | x² ≧ y} \ {0}.