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In this paper, we study genus $0$ equivariant relative Gromov-Witteninvariants of $\mathbb{P}^1$ whose corresponding relative stable maps aretotally ramified over one point. For fixed number of marked points, we showthat such invariants are piecewise polynomials in some parameter space. Theparameter space can then be divided into polynomial domains, called chambers.We determine the difference of polynomials between two neighboring chambers. Insome special chamber, which we called the totally negative chamber, we showthat such a polynomial can be expressed in a simple way. The chamber structurehere shares some similarities to that of double Hurwitz numbers.

Chamber structure for some equivariant relative Gromov-Witten invariants of $\mathbb{P}^1$ in genus $0$