2014/11/27 by Benoît Loridant, Loridant, Benoît
Computer Science · Materials Science · Mathematics · #11A63 #28A80 #54F65 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties #math.DS #msc:11A63 #msc:28A80 #msc:54F65 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1411.7544
44 pages
arxiv created 2014/11/27 · openalex publication_date 2014/11/27 · arxiv updated 2014/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the substitution σa,b defined by σa,b: · 1 · ↦ · \underbrace1… 1a2
· 2 · ↦ · \underbrace1… 1b3
· 3 · ↦ · 1 with a≥ b≥ 1. The shift dynamical system induced by σa,b is measure theoretically isomorphic to an exchange of three domains on a compact tile Ta,b with fractal boundary. We prove that Ta,b is homeomorphic to the closed disk iff 2b-a≤ 3. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a Hölder continuous parametrization Ca,b:\mathbbS1→∂ Ta,b of the boundary of Ta,b. As a by-product, this parametrization gives rise to an increasing sequence of polygonal approximations of ∂ Ta,b, whose vertices lye on ∂ Ta,b and have algebraic pre-images in the parametrization.