Curvature measures and fractals

Curvature measures and fractals

Curvature measures and fractals

Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ‘classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets F ⊆ R...

An Introduction to Fractals and Hausdorff Measures

It is possible for one to define fractals as those sets which have non-integral Hausdorff Dimension. This paper defines Hausdorff Dimension and rigorously introduces the necessary theory to prove that one such set is indeed a fractal. The first chapter includes a proof of Banach’s Fixed Point Theorem. The Hausdorff Metric is defined and it is mentioned how one produces ‘generators’ for creating...

Asymptotic degrees and measures on folded fractals

We study hyperbolic non-invertible maps f on fractal sets Λ of saddle type, preserving an equilibrium (Gibbs) measure μφ associated to an arbitrary Hölder potential φ. We prove a formula for a measure-theoretic asymptotic logarithmic degree of f |Λ with respect to μφ. This gives the average value of the logarithmic growth of the preimage counting functions of f|Λ, n ≥ 1, in the case when f |Λ i...

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ-reach can be estimated by the same curvature measures of the offset of a compact set K’ close to K in the Hausdorff sense. We show how these curvature measures can...