A New Approach to Find the Multi-Fractal Dimension of Multi-Fuzzy Fractal Attractor Sets Based on the Iterated Function System

In nature, objects are not single fractal sets but are a collection of complex multiple fractals that characterise the multifractal space, a generalisation of fractal space. While fractal space includes a fractal set, a multi-fractal space includes the union of fractals. A fuzzy fractal space is a fuzzy metric space and is an approach for the construction, analysis, and approximation of sets and images that exhibit fractal characteristics. The finite Cartesian product of fuzzy fractal spaces is called the multi-fuzzy fractal space. We propose in this paper, a theoretical proof to define the multi-fractal dimensions FD of a multi-fuzzy fractal attractor of n objects for the self-similar fractals sets A = frl Ai = (Ai, A2,...An) of the contraction mapping W** : Pi (n)(i=1) H(F (X-i)) -> > Pi (n)(i=1) H(F (X-i)) with contractivity factor r = max{r(i), i = 1, 2,... n} where H(F (X-i) is a fuzzy fractal space for each i = 1, 2,..., n; over a complete metric space (Pi (n)(i=1) H(F (X-i)), D*) then for all B-i that belong toH(F (X-i)), there exists B* belonging to (Pi (n)(i=1) H(F (X-i)) such that W** (B* = Pi (n)(i=1) B-i) Pi (n)(i=1) U(j=1)(n)i U-k=1(k(i,j)) omega(ij)*(K) U-j=1(n)= Wi(B*)). By supposing that M (t) = (1 Sigma (r(ij)*(k))(FD))(nxn) is the matrix 12X12 associated with the the contraction mapping omega(ij)*(k). with contraction factor r(ij)*(k)., for all i, j = 1, 2,..., n, for all k = 1, 2,..., k(i, j), for all t >= 0, and h (t) = det(M (t) I). Then, we prove that if there exists a FD such tat; h(FD) = 0, then FD is the multi fractal dimension for the multi fuzzy-fractal sets of IFS; and M(FD) has a fixed point in R-n.