Empirical multifractal moment measures and moment scaling functions of self-similar multifractals.

Let S-i: R-d --> R-d for i = 1, n be contracting similarities, and let (p(1),...... p(n)) be a probability vector. Let K and p be the self-similar set and the self-similar measure associated with (S-i,p(i))(i). For q is an element of R and r > 0, define the qth covering moment and the qth packing moment of p by

N-q(r) = inf(E xis an element ofE)Sigmamu(B(x,r))(q), M-q(r) sup(F) (xis an element ofF)Sigmamu(x, r))(q),

where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied, then it is well known that

(*) lim(rSE arrow0)log N-q(r)/-log r = lim(rSE arrow0) log M-q(r)/-log r = beta(q) for q is an element of R,

where beta(q) is defined by Sigma(i)p(i)(q)r(j)(beta(q)) = 1 (here r(i) denotes the Lipschitz constant of S-i). Assuming the OSC, we determine the exact rate of convergence in there exist multiplicatively periodic functions pi(q), Pi(q): (0, infinity) --> R such that

(**) N-q(r)/(r) + epsilon(r), M-q(r)/r(-beta(q)) = Pi(q)(r) + epsilon(r)

where epsilon (r) --> 0 as r SE arrow 0. As an application of we show that the empirical multifractal moment measures converges weakly:

1/Nq(r)(xis an element ofEr)Sigmamu(B(x,r))(q)delta(x) --> H-mu(q,beta(q)) K/H-mu(q,beta(q))(K) = P-mu(q,beta(q)) K/P-mu(q,beta(q))(K) weakly as rSE arrow0,

1/M-q(r)(xis an element ofFr)Sigmamu(B(x,r))(q)delta(x) --> H-mu(q,beta(q)) K/H-mu(q,beta(q))(K) = P-mu(q,beta(q)) K/P-mu(q,beta(q))(K) weakly as rSE arrow0,

where, for each positive r, E-r is a (suitable) minimal r-spanning subset of K and F, is a (suitable) maximal r-separated subset of K, and H-mu(q,beta(q)),o(q) and P-mu(q,beta(q))) are the multifractal Hausdorff measure and the multifractal packing measure, respectively.