Polynomial Curve Systems are Exponentially Decaying

Abstract

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 $f$: \[ N_{\mathcal{F}}(\eta) \le A \, N_{\mathcal{F}}(C) \, e^{-n \delta} + D, \qquad \forall n\ge 1, \] 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 $\mathcal{F}$, $\delta> 0$ is a constant depending only on $f$, $A, D > 0$ are constants depending only on $\mathcal{F}$ and $f$, and $\eta$ 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.