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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 $F\subseteq {ℝ}^d$ (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic...

In some recent work, fractal curvatures $C_k^f\left(F\right)$ and fractal curvature measures $C_k^f(F,·)$, $k=0,...,d$, have been determined for all self-similar sets $F$ in ${ℝ}^d$, for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations...

We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̅(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates $f^{\circ n}$ of a distance doubling map f exhibit “distance doubling behavior”. The results include well known statements for h related to the structure of the Mandelbrot set M. For h̅ they suggest some analogies to the structure of...