Curvature-direction measures of self-similar sets
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Curvature-direction measures of self-similar sets Tilman Johannes Bohl · Martina Zähle
Received: 18 November 2011 / Accepted: 28 November 2012 / Published online: 18 December 2012 © Springer Science+Business Media Dordrecht 2012
Abstract We obtain fractal Lipschitz–Killing curvature-direction measures for a large class of self-similar sets F in Rd . Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions. Keywords Self-similar set · Lipschitz–Killing curvature-direction measure · Fractal curvature measure · Minkowski content Mathematics Subject Classification (2000) Secondary: 28A78, 53C65
Primary: 28A80, 28A75, 37A99 ·
1 Introduction The “second order” anisotropic structure of self-similar sets F in Rd is studied by means of approximation with parallel sets F() of small distances . This leads to fractal curvaturedirection measures and their local “densities”. From the isotropic point of view this was first investigated in the pioneering work by Winter [24] (deterministic self-similar sets with
Both supported by grant DFG ZA 242/5-1. The first author has previously worked under the name Tilman Johannes Rothe. T. J. Bohl (B) · M. Zähle Friedrich Schiller University Jena, Jena, Germany e-mail: [email protected] M. Zähle e-mail: [email protected]
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Geom Dedicata (2013) 167:215–231
polyconvex parallel sets, curvature measures) and then Zähle in [28] (self-similar random sets with singular parallel sets, total curvatures), Winter and Zähle [26] (deterministic measure version for singular parallel sets), and Rataj and Zähle [20] (dynamical approach to curvatures measures and their local densities). The special case of the Minkowski content was treated earlier, e.g. Lapidus and Pomerance [13], and Falconer [4] (d = 1), Gatzouras [9] (selfsimilar random fractals for any d), Kesseböhmer and Kombrink [12] (d = 1 self-conformal sets), and in a very general context recently by Rataj and Winter [17]. In the present paper we extend these results to anisotropic quantities for the fractal sets (cf. the remark at the end of the paper). We mainly follow the dynamical approach from [20] and give a new and short proof for convergence of the corresponding measures under weaker assumptions, which considerably simplifies the former approaches. The classical geometric background are extensions of Federer’s curvature measures for sets of positive reach [6] regarding normal directions, the so-called curvature-direction measures or generalized curvature measures (cf. [21,27], and various
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