Hausdorff dimension, an extended non-negative real number associated with any metric space that generalizes the notion of the dimension of a real vector space
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For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ℝ.