Finite orbits in multivalued maps and Bernoulli convolutions

Finite orbits which intersect D have a network-like structure and can exist only at Perron parameters β. Their points are intersections of extended address curves, and can have finite or countable number of addresses, as found by Sidorov. For an uncountable number of parameters, the central point 12 has only two addresses. The intersection of periodic address curves can lead to singularities of the measures. We give examples which are not Pisot or Salem parameters.