垫片上自避路径的重整化群映射的推广所得到的二维映射的不动点的唯一性
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
摘要：
令$W(x,y) = a x^3 + b x^4 + f_5 x^5 + f_6 x^6 + (3 a x^2)^2 y + g_5 x^5 y + h_3 x^3 y^2 + h_4 x^4 y^2 + n_3 x^3 y^3 + a_{24} x^2 y^4 + a_{05} y^5 + a_{15} x y^5 + a_{06} y^6$，和$X=\frac{\partial W}{\partial x}$，$Y=\frac{\partial W}{\partial y}$，其中系数是非负常数，具有$a>0$，使得$X^{2}(x,x^{2})-Y(x,x^{2})$是具有非负系数的$x$的多项式。满足条件的二维映射$Φ: (x,y)\mapsto (X(x,y),Y(x,y))$的例子是三维和四维预垫片上的受限自回避路径的重整化群（RG）映射（变量的模变化）。证明了在不变集$\{(x,y)\in R^2\mid x^2\ge y\}\setminus\{0\}$中存在唯一的不动点$(x_f,y_f)$.
英文摘要：
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 $Φ: (x,y)\mapsto (X(x,y),Y(x,y))$ 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 $Φ$ in the invariant set $\{(x,y)\in R^2\mid x^2\ge y\}\setminus\{0\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
