Bernoulli Convolutions and 1D Dynamics

We describe a family $ϕ_λ$ 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 $ϕ_λ$ and give some numerical evidence to suggest values of $λ$ for which $ϕ_λ$ may be piecewise convex.

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