This thesis studies the absolute continuity of stationary measures. Given a finite set of measurable maps $S1,S2,\dots ,Sn$ on a measurable set $X$ and a probability vector $p1,p2,\dots ,pn$ we say that a probability measure $\nu$ on $X$ is stationary if

$\nu =\sum i=1npi\nu \circ Si-1.$

If $S1,\dots ,Sn$ are elements of PSL₂($R$) acting on X = P¹($R$), we get the notion of Furstenberg measures. If $S1,\dots ,Sn$ are similarities, affine maps, or conformal maps then $\nu$ is called a self-similar, self-affine, or self-conformal measure respectively. This thesis is concerned with Furstenberg measures and self-similar measures.

Two fundamental questions about stationary measures are what are their dimensions and when are they absolutely continuous. This thesis deals with the second one of these.

There are several classes of stationary measures which are known to be absolutely continuous for typical choices of parameters. For example Solomyak showed that for almost every $\lambda \in (1/2,1)$ the Bernoulli convolution with parameter $\lambda$ is absolutely continuous. This was extended by Shmerkin who showed that the exceptional set has Hausdorff dimension zero. However, despite much effort, there are relatively few known explicit examples of stationary measures which are absolutely continuous.

In this thesis we find sufficient conditions for self-similar measures and Furstenberg measures to be absolutely continuous. Using this we are able to give new examples.

The techniques we use are largely inspired by the techniques of Hochman and Varj'u though we introduce several new ingredients the most important of which is ``detail'' which is a quantitative way of measuring how smooth a measure is at a given scale.