Dynamics of relations associated with two core decompositions

Given an unshielded continuum K⊂C, the core decomposition DKLC of K with respect to local connectedness is known to exist [2]. Such a decomposition DKLC is monotone, locally connected under quotient topology, and refines every other monotone decomposition D′ which is locally connected under quotient topology. Let ∼ be the closed equivalence whose classes form the decomposition DKLC. Then ∼ contains a symmetric closed relation RK which requires (x, y) ∈ RK if and only if x and y lie in a prime end impression of C∖K. We also say that the relation RK respects prime end impressions. From Akin’s viewpoint of dynamical systems, the equivalence ∼ may be obtained as one of the limit relations of RK, through transfinite process. We will propose a direct approach to realize this limit relation in a concrete construction. More or less, such an approach builds the elements of DKLC “from below ” . If K is an unshielded disconnected compactum, the core decomposition DKFS of K with respect to the finitely suslinian property has been obtained in [3]. In this case, we consider a symmetric closed relation that “respects limit continua” and show that the same approach also works, in constructing DKFS from below.