Slow and fast convergence to local dimensions of self-similar measures

Let K and mu be the self-similar set and the self-similar measure associated with an iterated function system with probabilities (S-i, p(i))(i=1),...(N) satisfying the Open Set Condition. Let Sigma = {1,...,N}(N) denote the full shift space and let pi : Sigma --> K denote the natural projection. The (symbolic) local dimension of mu at w is an element of Sigma is defined by lim(n) log muK(w\n)/log diam K-w\n where K-w\n = S-w1 o...o S-wn (K) for w = w(1)w(2)... epsilon Sigma, and the (symbolic) multifractal spectrum of mu is defined by

f(s)(alpha) := dim pi{w epsilon Sigma \ lim(n) log muK(w\n)/log diam K-w\n = alpha}, alpha greater than or equal to 0,

where dim denotes the Hausdorff dimension. In this paper we study the speed with which the ratio log muK(w\n)/log diam K-w\n converges. In particular, we prove that for all (sufficiently large) speeds gamma, the set of points

{w epsilon Sigma \ lim(n)sup \log muK(w\n) - alpha log diam K-w\n\/rootn log log n = gamma}

for which the ration log muK(w\n)/log diam K-w\n converges to its limit with speed equal to gamma, has full dimension. i.e.

dim pi {w epsilon Sigma \ lim(n)sup \log muK(w\n) - alpha log diam K-w\n\/rootn log log n = gamma} = f(s)(alpha).

(C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.