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A combinatorial approach to products of Pisot substitutions. (English) Zbl 1378.37036
Summary: We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi-Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

MSC:
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B10 Symbolic dynamics
28A80 Fractals
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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References:
[1] DOI: 10.1090/S0002-9947-99-02360-0 · Zbl 0984.11008 · doi:10.1090/S0002-9947-99-02360-0
[2] DOI: 10.1007/s00605-008-0009-7 · Zbl 1190.11005 · doi:10.1007/s00605-008-0009-7
[3] DOI: 10.1007/BF01449880 · JFM 38.0262.01 · doi:10.1007/BF01449880
[4] Berthé, Unif. Distrib. Theory 7 pp 173– (2012)
[5] DOI: 10.1007/BF01637281 · Zbl 0543.10023 · doi:10.1007/BF01637281
[6] DOI: 10.1007/978-3-642-37067-0_10 · Zbl 1382.68246 · doi:10.1007/978-3-642-37067-0_10
[7] DOI: 10.1090/S0002-9947-02-03003-9 · Zbl 1042.37023 · doi:10.1090/S0002-9947-02-03003-9
[8] DOI: 10.1090/conm/385/07205 · doi:10.1090/conm/385/07205
[9] DOI: 10.1017/S0143385798100445 · Zbl 0915.58077 · doi:10.1017/S0143385798100445
[10] Berthé, Integers 11B (2011)
[11] DOI: 10.1515/crll.1868.69.29 · JFM 01.0062.01 · doi:10.1515/crll.1868.69.29
[12] Ito, Probability and Number Theory–Kanazawa 2005 pp 171– (2007)
[13] Barge, Bull. Soc. Math. France 130 pp 619– (2002) · Zbl 1028.37008 · doi:10.24033/bsmf.2433
[14] DOI: 10.1007/BF02771781 · Zbl 1143.37013 · doi:10.1007/BF02771781
[15] DOI: 10.3836/tjm/1270128186 · Zbl 0805.52011 · doi:10.3836/tjm/1270128186
[16] DOI: 10.1016/j.aim.2010.07.019 · Zbl 1219.37013 · doi:10.1016/j.aim.2010.07.019
[17] DOI: 10.5802/aif.2243 · Zbl 1194.11023 · doi:10.5802/aif.2243
[18] DOI: 10.2969/jmsj/05420283 · Zbl 1032.11033 · doi:10.2969/jmsj/05420283
[19] DOI: 10.1017/S0143385700007057 · Zbl 0814.68065 · doi:10.1017/S0143385700007057
[20] Akiyama, Algebraic Number Theory and Diophantine Analysis (Graz, 1998) pp 11– (2000)
[21] DOI: 10.1007/b13861 · Zbl 1014.11015 · doi:10.1007/b13861
[22] Akiyama, Number Theory and its Applications (Kyoto, 1997) pp 7– (1999)
[23] DOI: 10.1016/j.patcog.2008.11.003 · Zbl 1176.68180 · doi:10.1016/j.patcog.2008.11.003
[24] Siegel, Mém. Soc. Math. Fr. (N.S.) 118 pp 140– (2009)
[25] DOI: 10.1017/S0143385797084988 · Zbl 0884.58062 · doi:10.1017/S0143385797084988
[26] DOI: 10.1142/S0129054106004005 · Zbl 1096.68125 · doi:10.1142/S0129054106004005
[27] DOI: 10.4064/aa111-3-4 · Zbl 1051.11037 · doi:10.4064/aa111-3-4
[28] DOI: 10.1017/S0143385702001633 · Zbl 1042.37008 · doi:10.1017/S0143385702001633
[29] DOI: 10.1017/CBO9780511546716.010 · doi:10.1017/CBO9780511546716.010
[30] DOI: 10.1090/S0002-9947-01-02797-0 · Zbl 1142.37302 · doi:10.1090/S0002-9947-01-02797-0
[31] Schweiger, Multidimensional Continued Fractions (2000)
[32] DOI: 10.3934/dcds.2013.33.579 · Zbl 1291.37024 · doi:10.3934/dcds.2013.33.579
[33] Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory (1995) · Zbl 0819.11027
[34] Schweiger, The Metrical Theory of Jacobi–Perron Algorithm (1973) · Zbl 0287.10041 · doi:10.1007/BFb0059845
[35] DOI: 10.1017/CBO9780511777653 · doi:10.1017/CBO9780511777653
[36] DOI: 10.1007/BF02790261 · Zbl 0987.11013 · doi:10.1007/BF02790261
[37] Berthé, Integers 5 (2005)
[38] Brun, Treizième congrès des mathématiciens scandinaves, tenu à Helsinki 18–23 août 1957 pp 45– (1958)
[39] DOI: 10.1090/psapm/060/2078847 · doi:10.1090/psapm/060/2078847
[40] Brentjes, Multidimensional Continued Fraction Algorithms (1981)
[41] Rauzy, Bull. Soc. Math. France 110 pp 147– (1982) · Zbl 0522.10032 · doi:10.24033/bsmf.1957
[42] Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (2008) · Zbl 1172.37001
[43] DOI: 10.1090/S0002-9939-1978-0474415-8 · doi:10.1090/S0002-9939-1978-0474415-8
[44] DOI: 10.1007/978-3-642-11212-6 · Zbl 1225.11001 · doi:10.1007/978-3-642-11212-6
[45] DOI: 10.1353/ajm.2006.0037 · Zbl 1152.37011 · doi:10.1353/ajm.2006.0037
[46] Arnoux, Bull. Belg. Math. Soc. Simon Stevin 8 pp 181– (2001)
[47] Akiyama, Discrete Math. Theor. Comput. Sci. 7 pp 269– (2005)
[48] DOI: 10.5802/aif.2238 · Zbl 1138.37008 · doi:10.5802/aif.2238
[49] DOI: 10.1112/blms/bdq019 · Zbl 1211.11010 · doi:10.1112/blms/bdq019
[50] DOI: 10.1007/978-3-642-40579-2_9 · Zbl 1400.11075 · doi:10.1007/978-3-642-40579-2_9
[51] DOI: 10.3836/tjm/1270128497 · Zbl 0805.11056 · doi:10.3836/tjm/1270128497
[52] DOI: 10.1090/S0273-0979-98-00737-X · Zbl 0892.58019 · doi:10.1090/S0273-0979-98-00737-X
[53] DOI: 10.1016/S0022-314X(02)00076-8 · Zbl 1135.11326 · doi:10.1016/S0022-314X(02)00076-8
[54] DOI: 10.5802/aif.2079 · Zbl 1066.11032 · doi:10.5802/aif.2079
[55] Adler, Similarity of Automorphisms of the Torus (1970) · Zbl 0195.06104
[56] DOI: 10.1017/S0143385702001384 · Zbl 1031.11010 · doi:10.1017/S0143385702001384
[57] DOI: 10.4064/aa112-1-1 · Zbl 1060.11043 · doi:10.4064/aa112-1-1
[58] Arnoux, Bull. Soc. Math. France 119 pp 199– (1991) · Zbl 0789.28011 · doi:10.24033/bsmf.2164
[59] DOI: 10.1017/S0143385700003679 · Zbl 0625.28011 · doi:10.1017/S0143385700003679
[60] DOI: 10.1016/j.ejc.2014.01.009 · Zbl 1348.37027 · doi:10.1016/j.ejc.2014.01.009
[61] DOI: 10.4064/aa124-1-1 · Zbl 1116.28009 · doi:10.4064/aa124-1-1
[62] DOI: 10.1051/ita/2014008 · Zbl 1326.37007 · doi:10.1051/ita/2014008
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