MULTIFRACTAL STRUCTURE OF FRACTAL MEASURES

Abstract

In [HN], the authors showed that if 2 ≤ q ≤ m ≤ 2q − 2 then the set

E 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

jm probabilistic system p0 = p1 = ... = pm by the system pj = Cm/2 ,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].