Theoretical analysis reveals that quasisymmetric mappings cannot lower the Hausdorff dimension in fat and dense fractal percolations, indicating geometric rigidity in these random models.
We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.
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Anttila et al. (2026) studied this question.
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