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