  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.
