  We describe a family $\phi_{\lambda}$ of dynamical systems on the unit
interval which preserve Bernoulli convolutions. We show that if there are
parameter ranges for which these systems are piecewise convex, then the
corresponding Bernoulli convolution will be absolutely continuous with bounded
density. We study the systems $\phi_{\lambda}$ and give some numerical evidence
to suggest values of $\lambda$ for which $\phi_{\lambda}$ may be piecewise
convex.
