SOME OPEN QUESTIONS RELATED TO PROBABILITY, FRACTALS, WAVELETS AND TILING

Abstract

Let X0, X1, ... be a sequence of i.i.d. random variables each taking real values a1, ..., am with probability weights p1, ...,pm respectively. For 0 <ρ<1, let
S =
∞ n=0
ρnXn, (1)
and let μ be the probability measure induced by S. It is known thatμ is either purely singular or absolutely continuous. In general, we would like to know how the parametersρ, a1, ...,am and p,...,pm will affect the distribution type of μ. If μ is absolutely continuous, when will it have a derivative in Lp(1 ≤ p ≤∞)? When will it have a continuous derivative? If μ is singular, how is its multifractal structure? In this article we introduce some interesting problems arising from the study of above questions. We also explain how equation (1) is closely related to some models currently are being studied in fractals, wavelets and tiling.