2012/11/14 by Tilman Johannes Bohl, Bohl, Tilman Johannes
Mathematics · #28A75 #28A80 #37A30 #37A99 #53C65 #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #math.DS #math.MG #msc:28A75 #msc:28A78 #msc:28A80 #msc:37A30 #msc:37A99 #msc:53C65 #secondary: 28A78
paper · pdf · doi:10.48550/arxiv.1211.3421
34 pages, 2 figures
arxiv created 2012/11/14 · arxiv updated 2015/03/12
For self-similar fractals, the Minkowski content and fractal curvature have been introduced as a suitable limit of the geometric characteristics of its parallel sets, i.e., of uniformly thin coatings of the fractal. For some self-conformal sets, the surface and Minkowski contents are known to exist. Conformal iterated function systems are more flexible models than similarities. This work unifies and extends such results to general self-conformal sets in Rd. We prove the rescaled volume, surface area, and curvature of parallel sets converge in a Cesaro average sense of the limit. Fractal Lipschitz-Killing curvature-direction measures localize these limits to pairs consisting of a base point in the fractal and any of its normal directions. There is an integral formula. We assume only the popular Open Set Condition for the first-order geometry, and remove numerous geometric assumptions. For curvatures, we also assume regularity of the Euclidean distance function to the fractal if the ambient dimension exceeds three, and a uniform integrability condition to bound the curvature on "overlap sets". A limited converse shows the integrability condition is sharp. We discuss simpler, sufficient conditions. The main tools are from ergodic theory. Of independent interest is a multiplicative ergodic theorem: The distortion, how much an iterate of the conformal iterated function system deviates from its linearization, converges.