Normalizers of chains of discrete $p$-toral subgroups in compact Lie groups

In this paper we study the normalizer decomposition of a compact Lie group $G$ using the information of the fusion system $\mathcal{F}$ of $G$ on a maximal discrete $p$-toral subgroup. We prove that there is an injective map from the set of conjugacy classes of chains of $\mathcal{F}$-centric, $\mathcal{F}$-radical discrete $p$-toral subgroups to the set of conjugacy classes of chains of $p$-centric, $p$-stubborn continuous $p$-toral subgroups. The map is a bijection when $\pi_0(G)$ is a finite $p$-group. We also prove that the classifying space of the normalizer of a chain of discrete $p$-toral subgroups of $G$ is mod $p$ equivalent to the classifying space of the normalizer of the corresponding chain of $p$-toral subgroups.