As from the title, what is Bearnaise sauce thickened by? There are a few different ingredients in bearnaise sauce that helps in thickening such as egg yolks, but how is it thickened and what other ingredients are there that is considered as a thickening agent?
The thickening in Bearnaise, as in mayonnaise, is not so much in the ingredients as in the technique. These sauces get their thickness by being emulsified.
An emulsion is formed by rapidly mixing, whisking or blending two ingredients that shouldn't mix (oil and liquid). The emulsifier (egg yolk and often mustard in the case of mayo) stabilizes the emulsion (tiny oil droplets suspended in liquid) in much the same way detergents break up grease.
This wiki article goes into far more detail.