Yes gossip stores the petnames in your "People Lists > Followed" event (kind 3) when you publish that list.

Petnames can be strung together as paths.  If you read the bottom of NIP-02, you see an example `david.erin` which means "the person that erin calls david" (david within the subdomain of the person erin).  Dots in petnames would mess up such parsing/usage.  Gossip doesn't use petname paths anywhere, but we don't want to produce data in kind-3 follow lists that messes this up if other clients wanted to.