AI 中文总结
针对无限尖点树格$Y_q$的加权周期轨道Zeta函数，通过平稳Schur消元推导相关代数公式，计算得到Gurevich压力与无穷远处压力，证明强正回复性等性质。
AI 中文摘要
我们研究无限尖点树格$Y_q$的加权周期轨道Zeta函数，其中$q\ge2$为偶数，其商集为单侧梳状结构。全局欧拉乘积在系数层面失效，因为存在无穷多本原周期的长度为4。然而，有限有向边集上存在首返回行列式，平稳Schur消元给出根局部Zeta函数及其主导极点的代数公式。对两步重数势，我们计算得到Gurevich压力$P_G=2\log(q+1)$与无穷远处压力$P_\infty=\log(4q)$，由此得到强正回复性与指数局部轨道渐近性。高度阻尼转移算子为迹类算子。在减去显式积分压力抵消项后，逆Fredholm行列式具有局部一致有限部分，由收敛二重对数乘积表示且在高度变化下协变。
英文摘要
We study weighted periodic-orbit zeta functions for an infinitely cusped tree lattice $Y_q$, where $q\ge2$ is even and the quotient is a one-sided comb. The global Euler product fails coefficientwise because infinitely many primitive cycles have length four. A first-return determinant at a finite directed-edge set nevertheless exists, and stationary Schur elimination gives an algebraic formula for the root local zeta and its dominant poles. For the two-step multiplicity potential we compute the Gurevich pressure $P_G=2\log(q+1)$ and pressure at infinity $P_\infty=\log(4q)$, yielding strong positive recurrence and exponential local-orbit asymptotics. The height-damped transition operator is trace class. After subtraction of an explicit integrated-pressure counterterm, the inverse Fredholm determinant has a locally uniform finite part, expressed by a convergent dilogarithmic product and covariant under changes of height.
Comments34 pages, 1 figure, 3 tables
