Let λє(1/2,1) and consider the random variable ∑_k=1^∞±λ^k, where the signs are chosen randomly, independent and with equal probability. For λ=1/2 the distribution of this power series is the Lebesgue measure on (-1,1). It has been shown by Solomyak that for almost all λє(1/2,1), the distribution is absolutely continuous with respect to Lebesgue measure. Moreover Erdős has shown that if λ^-1 is a Pisot number, then the distribution is singular with respect to Lebesgue measure.

These results can be used to show the existence of absolutely continuous invariant measures for some piecewise hyperbolic maps, such as the fat bakers transformation.

In this talk we consider random power series of the form ∑_k=1^∞±λ_k where λ_k are independent and uniformly distributed in the interval (λ-ε,λ+ε). We show that if λ>1/2, then these power series have distributions in L₂ and that the L₂-norm does not grow faster than 1/√ε as ε vanishes. This is done by considering appropriate piecewise hyperbolic random dynamical systems. It will also be indicated that the method can be used to prove results for a big class of piecewise hyperbolic (random) dynamical systems.