Monodromy on the Central Hyperbolic Component of Polynomials with Just Two Distinct Critical Points

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For any given integers $d_0, d_1 \ge 1$, let $\mathcal{F}$ be the family of polynomial maps $f$ such that $f$ has a fixed point at the origin, and moreover has just two distinct critical points 1 and $c_f\neq 1$ of multiplicies $d_0$ and $d_1$, respectively. For the central hyperbolic component $\mathcal{H}$ of $\mathcal{F}$, a monodromy map on $\mathcal{H}$ is obtained by Branner-Hubbard deformations. We show that for any given $\lambda$ with $0 < |\lambda| < 1$ and for any given integer $n \ge 1$, the monodromy map transitively acts on the family of all polynomial maps $f\in \mathcal{H}$ with $f^{\prime}(0) = \lambda$ and $f^{\circ n}(c_f) = 1$.