Multiaffinity and entropy spectrum of self-affine fractal profiles

The entropy spectrum method is applied to self-affine fractal profiles. First, the profile created by a generalized multiaffine generator is decomposed into many subsets having their own topological entropies. The entropy spectrum and [Formula Presented] (the [Formula Presented]th Hurst exponent) of its profile is calculated exactly. For each subset, [Formula Presented] (divider dimension) and [Formula Presented] (box dimension) are also calculated. The relation [Formula Presented] is obtained for the remaining subset after infinite iteration of the generator. Next, the entropy spectrum of fractional Brownian motion (FBM) traces is examined and obtained as a point spectrum. This implies that a variety of lengths of segments in FBM traces is caused not by intrinsic inhomogeneity or mixing of the Hurst exponents but by only the trivial fluctuation. Namely, there are no fluctuations in singularity or in topological entropy. Finally, a real mountain range (the Hida mountains in Japan) is also analyzed by this method. Despite the profile of the Hida mountains having two Hurst exponents, the entropy spectrum of its profile becomes a point spectrum again.