2014/10/31 by Jonathan M. Fraser, JONATHAN M. FRASER, Jun Jie Miao +3
Computer Science · Mathematics · #Dimension (graph theory) #Effective dimension #Fractal #Hausdorff dimension #Mandelbrot set #Mathematical Dynamics and Fractals #Minkowski–Bouligand dimension #Packing dimension #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #math.DS #math.GN #math.MG #math.PR #msc:28A80 #msc:37C45 #msc:54E52 #msc:60J80 #msc:82B43
paper · pdf · doi:10.1017/etds.2016.64
published as Ergodic Theory and Dynamical Systems, 38, (2018), 982-1011 · 26 pages, 7 figures, v3 corrected error in the proof of Theorem 3.2 and sharpened results on exceptional sets
arxiv created 2015/05/01 · openalex created_date 2016/06/24 · openalex publication_date 2016/09/22 · arxiv updated 2018/04/26 · openalex updated_date 2026/08/05
We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.