Let $\mu$ be a positive finite Borel measure on the unit circle of the complex plane, and let $\newcommand{\cD}{{\mathcal{D}}} \cD(\mu)$ be the associated Dirichlet space. The Beurling-Deny capacity associated with $\cD(\mu)$ is denoted by $c_\mu$. Brown-Shields conjecture for $\cD (\mu)$ says that a function $f\in \cD (\mu)$ is cyclic for $\cD (\mu)$, meaning that $\{ pf: p \ \mbox{is a polynomial} \}$ is dense in $\cD (\mu)$, if and only if $f$ is outer and the boundary zeros set of $f$ is of $c_\mu-$ capacity zero. In this talk, we give a class of measures for which Brown-Shields conjecture holds. We also give an explicit description of all invariant subspaces for the shift operator for these measures. Our method is based on a study of the behavior of extremal functions for Dirichlet spaces.

This is joint work with Y. Elmadani and I. Labghail