Abstract
Abstract In this paper we want to give a new definition of fractal dimensions as small scale behavior of the q -energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show, that the 2- dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting on some Hilbert space. It will be proved that for ϕ, we haveand that where κ ± (2) are the upper and lower correlation dimensions of the spectral measure associated with Ψ and ϕ. A quantitative version of the RAGE theorem shall also be given.
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