Daniel Schnellmann: Absolutely continuous limit distributions of sums of point measures

In this talk I will present three results of my recent PhD thesis. In the first two results, we consider sequences of real numbers in the unit interval and study how they are distributed. The sequences in the first paper are given by the forward iterations of a point x ∈ [0,1] under a piecewise expanding map T_a:[0,1] → [0,1] depending on a parameter a contained in an interval I. Under the assumption that each T_a admits a unique absolutely continuous invariant probability measure μ_a and that some technical conditions are satisfied, we show that the distribution of the forward orbit T_a^j, j ≥ 1, is described by the distribution μ_a for Lebesgue almost every parameter a ∈ I. In the second result, we apply the ideas of the first paper to certain sequences which are equidistributed in the unit interval and give a geometrical proof of an old result by Koksma.

In the last result we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the to these Bernoulli convolutions associated.