Iterated Function Systems With Overlaps

In general it is difficult to study the multifractal structure of a self-similar measure when the associated iterated function system does not satisfy the open set condition. In this paper we will give two methods to deal with the overlapping situation. For the first method we make use of a transition matrix to calculate the Lp-scaling spectrum τ(p) of the measure. The second method depends on the Fourier transformation and the Ruelle operator; we use it to calculate the Sobolev exponent of the measure. We apply these two methods to study the m-th convolution of the Cantor measure, and also an interesting construction investigated recently by Kenyon [K] and Rao and Wen [RW]: S0(x) = 1/3 x, S1(x) = 1/3x+ λ/3, S2(x) = 1/3x + 2/3, with 0 < λ < 1, λ ∈ Q.