Abstract :
To characterize the geometry of a measure, its generalized dimensions dq have been introduced recently. The mathematically precise definition given by Falconer ["Fractal Geometry," 1990] turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range. A new definition is introduced, which is also based on box-counting, but which carries relevant information about μ for negative q also. In particular, the Legendre connection between generalized dimensions and the multifractal spectrum is established using large deviations and rigorous proofs are provided for the implicit formula giving the generalized dimensions of selfsimilar measures, which was until now known only for positive q.
