Connectedness and homogenization. Examples of fractal conductivity

A detailed study of the concept of $p$-connectedness is carried out; in particular, a criterion for the $p$-connectedness of two disjoint domains with Lipschitz boundaries and with fractal contact is formulated. New examples of open periodic sets with positive effective conductivity are constructed on the basis of this analysis. A new class of objects, elliptic operators in a Euclidean space with measure, is introduced; the corresponding concept of $p$-connectedness is introduced and a generalized theory of homogenization is developed.