- Adic Numbers And Continues Transition Between Dimensions
نویسنده
چکیده
Fractal measures of images of continuous maps from the set of p-adic numbers Q p 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 σ-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 Q p in C continuously dependent on " complex dimension " d ∈ C is obtained. This family is such that: 1) if d = 2(1), then the image of Q p is C (real axis in C); 2) the fractal measures coincide with the images of the Haar measure in Q p , 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 Q p are holomorphic in d, similarly to the dimensional regularization method in QFT. It is well-known that the Hausdorff measures (HM) are natural integral geometry characteristics for a wide class of sets in R d [6, 4, 7]. Therefore, contraction of a D-dimensional HM h D on D-dimensional rectifiable submanifolds is a measure of their areas [4] and, besides, there exist fractal subsets such that the HM contraction onto them for non-integer D is also nontrivial, i.e. is non-zero and (σ-) finite, and determines
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