Tetsuya Hattori
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
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ABSTRACT.  Let $W(x,y) = a x^3 + b x^4 + f_5 x^5 + f_6 x^6 + (3 a x^2)^2 y + g_5 x^5 y + h_3 x^3 y^2 + h_4 x^4 y^2 + n_3 x^3 y^3 + a_{24} x^2 y^4 + a_{05} y^5 + a_{15} x y^5 + a_{06} y^6$, 
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 $\{(x,y)\in R^2\mid x^2\ge y\}\setminus\{0\}$.