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

Abstract

Let $W(x,y)=a{x}^3+b{x}^4+f_5{x}^5+f_6{x}^6+(3a{x}^2{)}^{2}y+g_5{x}^{5}y+h_3{x}^3{y}^2+h_4{x}^4{y}^2+n_3{x}^3{y}^3+a_{24}{x}^2{y}^4+a_{05}{y}^5+a_{15}x{y}^5+a_{06}{y}^6$, and $X=\frac{\partial W}{\partial x}$, $Y=\frac{\partial W}{\partial y}$, where the coefficients are non-negative constants, with $a>0$, such that $X^2(x,x^2)-Y(x,x^2)$ is a polynomial of $x$ with non-negative coefficients. Examples of the 2 dimensional map $\Phi :(x,y)\mapsto (X(x,y),Y(x,y))$ 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 $(x_f,y_f)$ of $\Phi$ in the invariant set $\{(x,y)\in R^2\mid x^2\ge y\}\setminus \{0\}$.

LaTeX2e, 12 pages, no figures