I think an empty results list is just fine, when your page navigation is good.
An empty results list is consequent. It doesn't break the expectations (e.g. showing a page which doesn't look like the search page at all, like the "bummer no results" page) and conveys "here is an empty list".
The important part is your navigation. If there was a search, make sure the search field is visible above (use the same layoutfor full and empty result lists). If it was a navigation in a FAQ with categories, then make sure the needed breadcrumbs to navigate back are visible.
Do not redirect to some other page (this will break simple fixes like changing the mistyped term directly in the URL) and do not provide action buttons which are not clearly useful. When I found no answer in the FAQ category, I do not want to return to the homepage. Maybe I want to return to the FAQ index, most likely I want to return to the parent category. In search results, I most likely want to change the search term.