You are some what on the right track with aria-haspopup. And as @Menelion Elensúle said your arrow button has no content.
Let's say we do something like the iTunes store and have 1 link to a category and a dropdown for sub-categories... This is the proper markup
<a href="movies.html" clsas="acttion-button">Movies</a>
<button type="button" class="action-dropdown hide" aria-haspopup="true" aria-expanded="false" aria-controls="movies-menu">movies dropdown</button>
<ul id="movies-dropdown">
<li><a href="movies/adventure.html">Adventure</a></li>
...
</ul>
Use the class "hide" to visually hide the text. (Don't use display:none) And when the arrow is clicked change aria-expanded to true, and remember to switch it back to false when the dropdown is hidden.