Mathematics – Dynamical Systems
Scientific paper
2003-02-13
Mathematics
Dynamical Systems
24 pages, 3 figures
Scientific paper
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.
No associations
LandOfFree
Fractal Measures, p-Adic Numbers And Continues Transition Between Dimensions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Fractal Measures, p-Adic Numbers And Continues Transition Between Dimensions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fractal Measures, p-Adic Numbers And Continues Transition Between Dimensions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-333193