Boundary of the central hyperbolic component I: dynamical properties

We study the dynamics of polynomial maps on the boundary of the central hyperbolic component ${\mathcal{H}}_d$. We prove the local connectivity of Julia sets and a rigidity theorem for maps on the regular part of $\mathrm{\partial }{\mathcal{H}}_d$. Our proof is based on the construction of Fatou trees and employs the puzzle technique as a key methodological framework. These results are applicable to a larger class of maps for which the maximal Fatou trees equal the filled Julia sets.