Dual systems of algebraic iterated function systems. (English) Zbl 1294.28011
Let \((V, \Gamma)\) be a directed graph with vertex set \(V=[1, \dots, N]\) and edge set \(\Gamma\). Let \(F=(f_{\gamma}: {\mathbb R}^n \mapsto {\mathbb R}^n )_{\gamma\in\Gamma}\) be a family of contraction maps. The triple \((V, \Gamma, F)\) is called a graph-directed iterated function system (GIFS). Let \(\beta\) be an algebraic number. Denote \({\mathbb B}(\beta)\) the field generated by \(\beta\) and the rational field. A GIFS is called algebraic if there exists an algebraic number \(\beta>1\) such that \(f_{\gamma}(x) = (x+b_{\gamma})/\beta\) with \(b_{\gamma}\in {\mathbb B}(\beta)\) for each \(\gamma\). The main result of the paper is a construction of dual iterated function systems for algebraic GIFS. The paper contains results concerning feasible Pisot systems, Rauzy fractals and Rauzy-Thurston tilings.
Reviewer: Boris A. Kats (Kazan)
MSC:
| 28A80 | Fractals |
Keywords:
iterated function system; Rauzy fractal; self-similar tiling; Pisot spectrum conjecture; Rauzy-Thurston tilingsReferences:
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