Hausdorff dimension of reflected Bedford--McMullen carpets

Abstract

We study Bedford--McMullen type carpets whose selected grid rectangles may be reflected in one or both coordinates. The organizing principle is that the Hausdorff dimension is controlled by the entropy of the weak-coordinate projection. When this weak projection is separated, we obtain an explicit McMullen-type formula. This yields stability under arbitrary horizontal reflections and row-compatible weak reflections, and it gives several computable mixed-sign classes, including signed row-branch systems, interval-window systems and finite-block separated systems. We also explain why fully arbitrary weak-coordinate reflection patterns lead instead to a projection-entropy problem.