ar X iv : m at h / 05 02 27 7 v 1 [ m at h . D S ] 1 3 Fe b 20 05 INFINITE BERNOULLI CONVOLUTIONS AS AFFINE ITERATED FUNCTION SYSTEMS

We exploit the fact that the classical Bernoulli systems are con-tractive iterated function systems (IFS) of affine type to prove a number of properties of the infinite Bernoulli convolution measures ν λ. We develop and use a new duality notion for affine IFSs. This duality is based on a natural transfer operator R W , and on an associated random walk process Px. We show that the absolute-square of the Fourier transform of ν λ is the unique solution to a certain functional equation involving Px; and we use this in turn to establish a detailed harmonic analysis of the transfer operator R W , from which we derive our main results for the Bernoulli convolutions.