2013/10/09 by Bill Mance, Mance, Bill · 2 citations
Mathematics · #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1310.2377
Suppose that (P,Q) \∈ \ℕ2\ℕ \×\n\ℕ2\ℕ and x=E0.E1E2\⋯ is the P-Cantor series\nexpansion of x \∈ \ℝ. We define \ψP,Q(x):=\∑n=1^\∞\n frac \min(En,qn-1) q1 \⋯ qn. The functions \ψP,Q are used\nto construct many pathological examples of normal numbers. These constructions\nare used to give the complete containment relation between the sets of\nQ-normal, Q-ratio normal, and Q-distribution normal numbers and their\npairwise intersections for fully divergent Q that are infinite in limit. We\nanalyze the H "older continuity of \ψP,Q restricted to some judiciously\nchosen fractals. This allows us to compute the Hausdorff dimension of some sets\nof numbers defined through restrictions on their Cantor series expansions. In\nparticular, the main theorem of a paper by Y. Wang it et al.\n citeWangWenXi is improved.\n Properties of the functions \ψP,Q are also analyzed. Multifractal\nanalysis is given for a large class of these functions and continuity is fully\ncharacterized. We also study the behavior of \ψP,Q on both rational and\nirrational points, monotonicity, and bounded variation. For different classes\nof ergodic shift invariant Borel probability measures \μ1 and \μ2 on\n\ℕ2\ℕ, we study which of these properties \ψP,Q\nsatisfies for \μ1 \× \μ2-almost every (P,Q) \∈\n\ℕ2\ℕ \× \ℕ2\ℕ. Related classes of\nrandom fractals are also studied.\n