Polynomial Curve Systems are Exponentially Decaying

TL;DR

This paper proves that the pull back of a polynomial curve exponentially converges to the attractor, providing an alternative proof for the polynomial case of the finite global attractor conjecture by analyzing curve complexity reduction.

Contribution

It introduces the concepts of quick returns and barrier lakes to analyze curve dynamics, showing exponential complexity decay under polynomial pull back.

Findings

Curve complexity decreases exponentially under polynomial pull back

Provides an alternative proof for the polynomial case of the attractor conjecture

Introduces new combinatorial tools for analyzing polynomial curve dynamics

Abstract

The finite global attractor conjecture for the polynomial case was recently solved by using tree lifting algorithm [1]. It remains unclear how fast the pull back of a curve by a polynomial converges to the attractor. In the paper we introduce the ideas of quick returns and barrier lakes to analyze the combinatorial models of curves. These allow us to settle the question by showing that the complexity of a curve is exponentially decreased under the iteration of the pull back by a polynomial, and therefore, the pull back of a curve is exponentially contracted to the attractor. In particular, it gives an alternative proof of the finite global attractor conjecture for the polynomial case.