Univoque numbers and automatic sequences. (English) Zbl 1220.68075
Barral, Julien (ed.) et al., Recent developments in fractals and related fields. Based on the international conference on fractals and related fields, Monastir, Tunisia, September 2007 held in honor of Jacques Peyrière. Boston, MA: Birkhäuser (ISBN 978-0-8176-4887-9/hbk; 978-0-8176-4888-6/ebook). Applied and Numerical Harmonic Analysis, 383-392 (2010).
Summary: A set of binary sequences related to the iteration of unimodal continuous functions of the interval \([0,1]\) appears in a 1982–1983 work of Cosnard and the first author. An almost identical set of binary sequences occurs in a 1990 paper by Erdős, Joó, and Komornik; it consists of expansions of 1 in univoque bases \(\beta\) in \((1,2)\) (the base \(\beta\) is univoque if 1 admits a unique \(\beta\)-expansion). We generalize a result of the second author and Niu by proving, using the 1982–1983 results, that a large class of Thue-Morse-like sequences belong to these sets of binary sequences. The case of alphabets of size larger than 2 yields similar results.
For the entire collection see [Zbl 1195.28001].
For the entire collection see [Zbl 1195.28001].
MSC:
| 68R15 | Combinatorics on words |
| 11B83 | Special sequences and polynomials |
| 11B85 | Automata sequences |
| 37E05 | Dynamical systems involving maps of the interval |
Keywords:
iteration of unimodal continuous functions; univoque bases; Thue-Morse-like sequences; binary sequencesReferences:
| [1] | Allouche, JP, Théorie des Nombres et Automates (1983), Bordeaux: Thèse d’État, Bordeaux |
| [2] | Allouche, J-P, A note on univoque self-Sturmian numbers, RAIRO, Info. Théo. Appl., 42, 659-662 (2008) · Zbl 1197.11029 · doi:10.1051/ita:2007058 |
| [3] | Allouche, J-P; Clarke, M.; Sidorov, N., Periodic unique beta-expansions: the Sharkovskiĭ ordering, Ergod. Th. Dyn. Syst., 29, 1055-1074 (2009) · Zbl 1279.11009 · doi:10.1017/S0143385708000746 |
| [4] | Allouche, J-P; Cosnard, M., Itérations de fonctions unimodales et suites engendrées par automates, C.R.Acad.Sci.Paris, Sér.1, 296, 159-162 (1983) · Zbl 0547.58027 |
| [5] | Allouche, J-P; Cosnard, M., The Komornik-Loreti constant is transcendental, Am. Math. Mon., 107, 448-449 (2000) · Zbl 0997.11052 · doi:10.2307/2695302 |
| [6] | Allouche, J-P; Cosnard, M., Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set, Acta Math. Hung., 91, 325-332 (2001) · Zbl 1012.11007 · doi:10.1023/A:1010667918943 |
| [7] | Allouche, J-P; Frougny, C., Univoque numbers and an avatar of Thue-Morse, Acta Arith., 136, 319-329 (2009) · Zbl 1168.11002 · doi:10.4064/aa136-4-2 |
| [8] | Allouche, J-P; Frougny, C.; Hare, K., On univoque Pisot numbers, Math. Comp., 76, 1639-1660 (2007) · Zbl 1182.11051 · doi:10.1090/S0025-5718-07-01961-8 |
| [9] | Allouche, J-P; Rampersad, N.; Shallit, J., Periodicity, repetitions, and orbits of an automatic sequence, Theoret. Comput. Sci., 410, 2795-2803 (2009) · Zbl 1173.68044 · doi:10.1016/j.tcs.2009.02.006 |
| [10] | Allouche, J-P; Shallit, J.; Ding, C.; Helleseth, T.; Niederreiter, H., The ubiquitous Prouhet-Thue-Morse sequence, Sequences and their applications, Proceedings of SETA’98, 1-16 (1999), London: Springer, London · Zbl 1005.11005 |
| [11] | Allouche, J-P; Shallit, J., Automatic sequences. Theory, applications, generalizations (2003), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1086.11015 · doi:10.1017/CBO9780511546563 |
| [12] | Astudillo, R., On a class of Thue-Morse type sequences, J. Integer Seq., 6 (2003) · Zbl 1074.11016 |
| [13] | Cosnard, M., Étude de la classification topologique des fonctions unimodales, Ann. Inst. Fourier, 35, 59-77 (1985) · Zbl 0569.58004 |
| [14] | Daróczy, Z.; Kátai, I., Univoque sequences, Publ. Math. Debrecen, 42, 397-407 (1993) · Zbl 0809.11008 |
| [15] | Daróczy, Z.; Kátai, I., On the structure of univoque numbers, Publ. Math. Debrecen, 46, 385-408 (1995) · Zbl 0874.11013 |
| [16] | Dekking, FM, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheor. Verw. Geb., 41, 221-239 (1978) · Zbl 0348.54034 · doi:10.1007/BF00534241 |
| [17] | de Vries, M., A property of algebraic univoque numbers, Acta Math. Hungar., 119, 57-62 (2008) · Zbl 1164.11008 · doi:10.1007/s10474-007-6252-x |
| [18] | de Vries, M.; Komornik, V., Unique expansions of real numbers, Adv. Math., 221, 390-427 (2009) · Zbl 1166.11007 · doi:10.1016/j.aim.2008.12.008 |
| [19] | Doche, C., On the real roots of generalized Thue-Morse polynomials, Acta Arith., 99, 309-319 (2001) · Zbl 1020.11017 · doi:10.4064/aa99-4-1 |
| [20] | Erdős, P.; Joó, I.; Komornik, V., Characterization of the unique expansions \(1 ={ \sum \nolimits }_{i=1}^{\infty }{q}^{-{n}_i}\) , and related problems, Bull. Soc. Math. Fr., 118, 377-390 (1990) · Zbl 0721.11005 |
| [21] | Frid, A., Overlap-free symmetric D0L words, Discrete Math. Theoret. Comput. Sci., 4, 357-362 (2001) · Zbl 0990.68104 |
| [22] | Komornik, V.; Loreti, P., Unique developments in non-integer bases, Am. Math. Mon., 105, 636-639 (1998) · Zbl 0918.11006 · doi:10.2307/2589246 |
| [23] | Komornik, V.; Loreti, P., Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar., 44, 197-218 (2002) · Zbl 1017.11008 · doi:10.1023/A:1019696514372 |
| [24] | Komornik, V.; Loreti, P., On the topological structure of univoque sets, J. Number Theory, 122, 157-183 (2007) · Zbl 1111.11005 · doi:10.1016/j.jnt.2006.04.006 |
| [25] | Niu, M.; Wen, Z-X, A property of m-tuplings Morse sequence, Wuhan Univ. J. Nat. Sci., 11, 473-476 (2006) · Zbl 1155.11311 · doi:10.1007/BF02836646 |
| [26] | Shallit, J., A generalization of automatic sequences, Theoret. Comput. Sci., 61, 1-16 (1988) · Zbl 0662.68052 · doi:10.1016/0304-3975(88)90103-X |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
