Generalized fractal dimensions on the negative axis for compactly supported measures

We study generalized fractal dimensions of measures, called the Hentschel–Procaccia dimensions and the generalized Rényi dimensions. We consider compactly supported Borel measures with finite total mass on a complete separable metric space. More precisely we discuss in great generality finiteness and equality of the different lower and upper dimensions for negative values of their argument q . In particular we do not assume that the measure satisfies to the so called “doubling condition”. A key tool in our analysis is, given a measure μ , the function g ( ε ), ε > 0, defined as the infimum over all points x in the support of μ of the quantity μ ( B ( x , ε )), where B ( x , ε ) is the ball centered at x and of radius ε . We provide counter examples to show the optimality of some criteria for finiteness and equality of the dimensions. We also relate this work to quantum dynamics. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)