Equivalence of A-approximate continuity for self-adjoint expansive linear maps

Let A:Rd→Rd,d⩾1, be an expansive linear map. The notion of A-approximate continuity was recently used to give a characterization of scaling functions in a multiresolution analysis (MRA). The definition of A-approximate continuity at a point x – or, equivalently, the definition of the family of sets having x as point of A-density – depend on the expansive linear map A. The aim of the present paper is to characterize those self-adjoint expansive linear maps A1,A2:Rd→Rd for which the respective concepts of Aμ-approximate continuity (μ=1,2) coincide. These we apply to analyze the equivalence among dilation matrices for a construction of systems of MRA. In particular, we give a full description for the equivalence class of the dyadic dilation matrix among all self-adjoint expansive maps. If the so-called “four exponentials conjecture” of algebraic number theory holds true, then a similar full description follows even for general self-adjoint expansive linear maps, too.