Leading monomials of escape regions

This chapter considers a leading monomial vector, which uniquely determines the escape region and within which is encoded important information about the limiting behavior of the periodic critical orbit. Thus, for each p ≥ 1, this chapter considers the affine algebraic curve Sₚ formed by all monic and centered cubic polynomials with a marked critical point which has period p under iterations. Each unbounded hyperbolic component of Sₚ, called an escape region, has an associated vector of leading monomials which encodes the asymptotic behavior of the periodic critical orbit. The chapter shows that this vector determines the escape region, giving a positive answer to a question posed by Bonifant and Milnor.