We proved closed/dense dichotomy of $\operatorname{PGL}_2(\mathbb{Q}_p)$-orbit closures in the renormalized frame bundle of a $p$-adic homogeneous space of infinite volume. Our result is a generalization of Ratner’s theorem and the result of McMullen, Mohammadi, and Oh in 2017 into non-Archimedean local fields.

Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_p$. Our homogeneous space is a quotient space of $\operatorname{\mathbb{K}}$ by a certain class of Schottky subgroups. Using the main tools of McMullen, Mohammadi, and Oh, we introduced the necessary properties of Schottky subgroups and used the Bruhat-Tits tree $\operatorname{PGL}_2$. In this talk, we introduce the highly-branched Schottky subgroups and steps for the proof of the main theorem.

This is a joint work with Seonhee Lim.