Fractal Measures, p-Adic Numbers And Continues Transition Between Dimensions

Abstract

Fractal measures of images of continuous maps from the set of p-adic numbers Qp into complex plane C are analyzed. Examples of "anomalous" fractals, i.e. the sets where the D-dimensional Hausdorff measures (HM) are trivial, i.e. either zero, or sigma-infinite (D is the Hausdorff dimension (HD) of this set) are presented. Using the Caratheodory construction, the generalized scale-covariant HM (GHM) being non-trivial on such fractals are constructed. In particular, we present an example of 0-fractal, the continuum with HD=0 and nontrivial GHM invariant w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains in R^n, the formula for the change of variables for GHM is obtained. The family of continuous maps Qp in C continuously dependent on "complex dimension" d in C is obtained. This family is such that: 1) if d=2(1), then the image of Qp is C (real axis in C); 2) the fractal measures coincide with the images of the Haar measure in Qp, and at d=2(1) they also coincide with the flat (linear) Lebesgue measure; 3) integrals of entire functions over the fractal measures of images for any compact set in Qp are holomorphic in d, similarly to the dimensional regularization method in QFT.