Fractal wavelet dimensions and localization

In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-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 in some Hilbert space ?. It will be proved that for φ, ψ∈? we have $$mathop {lim inf }limits_{T to infty } frac{{log int_0^T {domega left| {leftlangle {psi left| {e^{ - iAomega } } right.phi } rightrangle } right|^2 } }}{{log T}} = - kappa ^ + (2)$$ and that $$mathop {lim sup }limits_{T to infty } frac{{log int_0^T {domega left| {leftlangle {psi left| {e^{ - iAomega } } right.phi } rightrangle } right|^2 } }}{{log T}} = - kappa ^ - (2),$$ wherek ^±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ?. A quantitative version of the RAGE theorem shall also be given.