Polynomial Curve Systems are Exponentially Decaying

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.