On a class of Rauzy fractals without the finiteness property. (English) Zbl 1462.11026
Summary: We present some topological and arithmetical aspects of a class of Rauzy fractals \(\mathcal{R}_{a,b}\) related to the polynomials of the form \(P_{a,b}(x)=x^3-ax^2-bx-1\), where \(a\) and \(b\) are integers satisfying \(-a+1 \leq b \leq -2\). This class has the property that \(0\) lies on the boundary of \(\mathcal{R}_{a,b}\). We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of \(\mathcal{R}_{a,b}\) in the tiling it generates. Furthermore, we prove that if \(2a+3b+4 \leq 0\) then \(\mathcal{R}_{a,b}\) is not homeomorphic to a topological disk. We also show that the boundary of the set \(\mathcal{R}_{3,-2}\) is generated by two infinite iterated function systems.
MSC:
| 11B85 | Automata sequences |
| 28A80 | Fractals |
| 37B10 | Symbolic dynamics |
| 52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |
References:
| [1] | S. Akiyama: Cubic Pisot units with finite beta expansions; in Algebraic Number Theory and Diophantine Analysis 2000, 11-26. · Zbl 1001.11038 |
| [2] | P. Arnoux, V. Berthé, H. Ei and S. Ito: Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions; in Discrete Math. Theor. Comput. Sci. AA 2001, 59-78. · Zbl 1017.68147 |
| [3] | P. Arnoux, V. Berthé and S. Ito: Discrete planes, \( \mathbb{Z}^2\)-actions, Jacobi-Perron algorithm and substitutions, Ann. Inst. Fourier (Grenoble) 52 (2002), 305-349. · Zbl 1017.11006 |
| [4] | P. Arnoux and S. Ito: Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181-207. · Zbl 1007.37001 |
| [5] | P. Arnoux, S. Ito and Y. Sano: Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math. 83 (2001), 183-206. · Zbl 0987.11013 · doi:10.1007/BF02790261 |
| [6] | P. Arnoux and G. Rauzy: Représentation géométrique des suites de complexité \(2n+1\), Bull. Soc. Math. France 119 (1991), 101-117. · Zbl 0789.28011 · doi:10.24033/bsmf.2164 |
| [7] | J. Bastos, A. Messaoudi, T. Rodrigues and D. Smania: A class of cubic Rauzy fractals, Theoret. Comput. Sci. 588 (2015), 114-130. · Zbl 1332.28012 · doi:10.1016/j.tcs.2015.04.007 |
| [8] | V. Berthé and A. Siegel: Purely periodic beta-expansions in the non-unit case, J. Number Theory 127 (2007), 153-172. · Zbl 1197.11139 · doi:10.1016/j.jnt.2007.07.005 |
| [9] | V. Canterini: Connectedness of geometric representation of substitutions of Pisot type, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 77-89. · Zbl 1031.37015 |
| [10] | V. Canterini and A. Siegel: Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144. · Zbl 1142.37302 |
| [11] | N. Chekhova, P. Hubert and A. Messaoudi: Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux 13 (2001), 371-394. · Zbl 1038.37010 |
| [12] | C. Frougny: Number representation and finite automata; in Topics in Symbolic Dynamics and Applications, London Math. Soc. Lectures Note Ser. 279, 2000, 207-228. · Zbl 0976.11003 |
| [13] | C. Holton and L. Zamboni: Geometric realizations of substitutions, Bull. Soc. Math. France 126 (1998), 149-179. · Zbl 0931.11004 |
| [14] | P. Hubert and A. Messaoudi: Best simultaneous diophantine approximation of Pisot numbers and Rauzy fractals, Acta Arith. 124 (2006), 1-15. · Zbl 1116.28009 · doi:10.4064/aa124-1-1 |
| [15] | R. Kenyon and A. Vershik: Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems 18 (1998), 357-372. · Zbl 0915.58077 |
| [16] | J.C. Lagarias, H.A. Porta and K.B. Stolarsky: Asymmetric tent map expansions. I. Eventually periodic points, J. London Math. Soc. 47 (1993), 542-556. · Zbl 0809.11041 |
| [17] | B. Loridant: Topological properties of a class of cubic Rauzy fractals, Osaka J. Math. 53 (2016), 161-219. · Zbl 1418.37031 |
| [18] | B. Loridant, A. Messaoudi, J. Thuswaldner and P. Surer: Tilings induced by a class of Rauzy fractals, Theoret. Comput. Sci. 477 (2013), 6-31. · Zbl 1271.37014 |
| [19] | A. Messaoudi: Propriétés arithmétiques et dynamiques du fractal de Rauzy, J. Théor. Nombres Bordeaux 10 (1998), 135-162. · Zbl 0918.11048 |
| [20] | A. Messaoudi: Frontière du fractal de Rauzy et système de numération complexe, Acta Arith. 95 (2000), 195-224. · Zbl 0968.28005 |
| [21] | A. Messaoudi: Propriétés arithmétiques et topologiques d’une classe d’ensembles fractales, Acta Arith. 121 (2006), 341-366. · Zbl 1104.28003 · doi:10.4064/aa121-4-5 |
| [22] | W. Parry: On the beta-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. · Zbl 0099.28103 |
| [23] | G.A. Pavani: Fractais de Rauzy, autômatos e fraçoes contínuas (Portuguese). Rauzy Fractals, Automata, and Continued Fractions, PhD Thesis, Saõ Paulo State University (2015), available at http://hdl.handle.net/11449/127805. |
| [24] | B. Praggastis: Numeration systems and Markov partitions from self-similar tilings, Trans. Amer. Math. Soc. 351 (1999), 3315-3349. · Zbl 0984.11008 |
| [25] | G. Rauzy: Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147-178. · Zbl 0522.10032 |
| [26] | A. Rényi: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. · Zbl 0079.08901 |
| [27] | S. Ito and M. Kimura: On Rauzy fractal, Japan J. Indust. Appl. Math. 8 (1991), 461-486. · Zbl 0734.28010 |
| [28] | K. Schmidt: On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269-278. · Zbl 0494.10040 |
| [29] | A. Siegel and J. Thuswaldner: Topological properties of Rauzy fractals, Mém. Soc. Math. Fr. 118 (2009), 144pp. · Zbl 1229.28021 |
| [30] | V. Sirvent and Y. Wang: Self-affine tiling via substitution dynamical systems and Rauzy fractals, Pacific J. Math. 206 (2002), 465-485. · Zbl 1048.37015 |
| [31] | C.J. Smyth: There are only eleven Special Pisot numbers, Bull. London Math. Soc. 31 (1999), 1-5. · Zbl 0931.11042 |
| [32] | W. Thurston: Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes 1 (1989). |
| [33] | J.M. Thuswaldner: Unimodular Pisot substitutions and their associated tiles, J. Théor. Nombres Bordeaux 18 (2006), 487-536. · Zbl 1161.37016 |
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