MULTIFRACTAL STRUCTURE OF FRACTAL MEASURES

Abstract

In [HN], the authors showed that if 2 ≤ q ≤m ≤ 2q−2 then the setE of the attainable local dimensions of fractal measure µ is an interval. In this paper we will prove that this result is not true if we replace the probabilistic system p0 = p1 = ...= pm by the system pj = Cj m/2m,j= 0,1,...,m. More precisely, the set E has an isolated point. Hence the multifractal formalism fails in this case. The special of our case when q = 3, the results was obtained earlier in [HL].