Hausdorff dimension of reflected Bedford--McMullen carpets

TL;DR

This paper investigates the Hausdorff dimension of reflected Bedford--McMullen carpets, providing explicit formulas and stability results for various reflection patterns based on entropy of projections.

Contribution

It introduces a McMullen-type formula for carpets with reflected grid rectangles and analyzes stability under different reflection symmetries.

Findings

Explicit dimension formulas for separated weak projections.

Stability under horizontal and row-compatible reflections.

Identification of classes with computable mixed signs.

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.