Invariance of the Hausdorff Dimension of McMullen-Bedford Carpets under Coordinate Reflections

TL;DR

This work extends the classical McMullen-Bedford carpets to include coordinate reflections and proves that the Hausdorff dimension is unchanged by these orientation-reversing maps. The authors derive both the upper and lower bounds using established fractal-geometry techniques: a standard covering argument yields the upper bound, while a dimension-maximizing Bernoulli measure combined with the Ledrappier-Young formula provides the lower bound. A key insight is that fiber entropies, which govern the Ledrappier-Young dimension, are invariant under reflections, ensuring the classical dimension formula $s = \frac{1}{\log m} \log (\sum_j t_j^{\log_n m})$ remains valid. Consequently, the dimension is dictated by the combinatorial fiber structure rather than the orientation of the grid, demonstrating robustness of the invariant under local isometries and extending Bedford-McMullen carpets to orientation-reversing systems.

Abstract

We analyze a generalization of the self-affine carpets of Bedford and McMullen where the defining iterated function system includes coordinate reflections. We show that the Hausdorff dimension is invariant under such reflections. The upper bound follows from the standard covering argument using approximate squares, while the lower bound is established by constructing a dimension-maximizing Bernoulli measure and applying the Ledrappier-Young formula. The key to the proof is the observation that the fiber entropies determining the dimension are invariant under the action of the reflection group.

Figures (1)

• Figure 1: The first iteration of a generalized McMullen-Bedford IFS with $n=3$ and $m=4$. The orientation of the letter 'F' illustrates the coordinate reflections $(\sigma_x, \sigma_y)$ applied in each selected rectangle.

Theorems & Definitions (4)

• Theorem 1.1
• Lemma 2.1: Invariance of Entropies
• proof
• Remark 3.1