A new pointwise invariant, the Point-Cross Dimension, quantifies the cumulative weighted complexity of independent directional channels through a single germ, separating directionality from isotropic dispersion.
Tangents of invariant sets
2 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
We study the fine scaling properties of sets satisfying various weak forms of invariance. For general attractors of possibly overlapping bi-Lipschitz iterated function systems, we establish that the Assouad dimension is given by the Hausdorff dimension of a tangent at some point in the attractor. Under the additional assumption of self-conformality, we moreover prove that this property holds for a subset of full Hausdorff dimension.
fields
math.MG 2years
2026 2representative citing papers
Marstrand's projection theorem fails for the Assouad spectrum and quasi-Assouad dimension, with new almost-sure lower bounds from capacity profiles and upper bounds from tube-counting for planar sets.
citing papers explorer
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Point-dimension theory (part II): The point-cross dimension
A new pointwise invariant, the Point-Cross Dimension, quantifies the cumulative weighted complexity of independent directional channels through a single germ, separating directionality from isotropic dispersion.
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On the Marstrand projection theorem for the Assouad spectrum
Marstrand's projection theorem fails for the Assouad spectrum and quasi-Assouad dimension, with new almost-sure lower bounds from capacity profiles and upper bounds from tube-counting for planar sets.