多项式曲线系统是指数衰减的
Polynomial Curve Systems are Exponentially Decaying
AI总结：
本文引入“快速返回”和“屏障湖”概念分析多项式曲线组合模型，证明曲线复杂度在多项式拉回迭代下指数衰减，从而定量证明了多项式情形下有限全局吸引子猜想。
摘要：
自Belk等人[4]的工作以来，多项式曲线系统存在有限全局吸引子已为人所知。然而，除双曲情形外，曲线在多项式下拉回的收敛速率仍不清楚。本文引入$ extit{快速返回}$和$ extit{屏障湖}$的概念来分析曲线的组合模型。这些概念使我们能够证明：如果一定次数的连续拉回未能按确定比例降低曲线的复杂度，则该曲线允许一个$ extit{厚}$-$ extit{薄}$ $ extit{分解}$：曲线的大部分被组织成有限个不相交的环带，其核心曲线具有有界同伦类型。在这种情况下，我们可以证明一定次数的连续拉回必然按确定因子降低曲线的复杂度。这意味着曲线$C$的复杂度在多项式$f$的拉回迭代下指数衰减：\[ N_{\mathcal{F}}(η) \le A \, N_{\mathcal{F}}(C) \, e^{-n δ} + D, \qquad orall n\ge 1, \] 其中$\mathcal{F}$是可容许分离弧族，$N_{\mathcal{F}}(\cdot)$表示曲线在其同伦类中与$\mathcal{F}$中弧的最小相交数，$δ>0$是仅依赖于$f$的常数，$A, D > 0$是仅依赖于$\mathcal{F}$和$f$的常数，$η$是$f^{-n}(C)$的任意分支。因此，曲线的拉回指数收缩到吸引子。特别地，这为多项式情形下的有限全局吸引子猜想提供了定量证明。
英文摘要：
The existence of a finite global attractor for polynomial curve system has been known since the work of Belk et al. [5]. However, except in the hyperbolic case, the rate at which the pullback of a curve under a polynomial converges to the attractor remained unclear. In this paper, we introduce the notions of $\textit{quick returns}$ and $\textit{barrier lakes}$ to analyze the combinatorial models of curves. These concepts allow us to show that if a certain number of successive pullbacks do not decrease the complexity of the curve by a definite proportion, then the curve admits a $\textit{thick}$-$\textit{thin}$ $\textit{decomposition}$: most of the curve is organized into finitely many disjoint annuli whose core curves have bounded homotopy type. In this case, we can show that some number of successive pullbacks must decrease the complexity of the curve by a definite factor. This implies that the complexity of a curve $C$ decreases exponentially under iteration of the pullback by a polynomial $f$: \[ N_{\mathcal{F}}(η) \le A \, N_{\mathcal{F}}(C) \, e^{-n δ} + D, \qquad \forall n\ge 1, \] where $\mathcal{F}$ is an admissible family of separation arcs, $N_{\mathcal{F}}(\cdot)$ denotes the minimal intersection number of the curves in its homotopy class and the arcs in $\mathcal{F}$, $δ> 0$ is a constant depending only on $f$, $A, D > 0$ are constants depending only on $\mathcal{F}$ and $f$, and $η$ is any component of $f^{-n}(C)$. Consequently, the pullback of a curve contracts exponentially to the attractor. In particular, this provides a quantitative proof of the finite global attractor conjecture for the polynomial case.
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