A fundamental and challenging problem in complex dynamics is to study the topological structure of high-dimensional hyperbolic component boundaries. In this talk, we will discuss the capture hyperbolic components in polynomial families defined by periodic critical relations. We show that the boundary of a capture hyperbolic component is homeomorphic to a high-dimensional sphere. Moreover, a relation for Hausdorff dimensions between the parameter space and the dynamical space will be mentioned. This is a joint work with Xiaoguang Wang and Yongcheng Yin.
