Bernoulli convolutions and 1D dynamics
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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.