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On multiplicatively dependent linear numeration systems, and periodic points. (English) Zbl 1044.11004
The author proves that the conversion between two linear numeration systems whose characteristic polynomials are, respectively, equal to the minimal polynomials of two multiplicatively dependent Pisot numbers can be computed by a finite automaton.
MSC:
11A67Representation systems for integers and rationals
68Q45Formal languages and automata
37B10Symbolic dynamics
68R15Combinatorics on words
11R06Special algebraic numbers
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References:
[1] M.-J. Bertin , A. Decomps-Guilloux , M. Grandet-Hugot , M. Pathiaux-Delefosse and J.-P. Schreiber , Pisot and Salem numbers . Birkhäuser ( 1992 ). Zbl 0772.11041 · Zbl 0772.11041
[2] A. Bertrand , Développements en base de Pisot et répartition modulo 1 . C. R. Acad. Sci. Paris 285 ( 1977 ) 419 - 421 . MR 447134 | Zbl 0362.10040 · Zbl 0362.10040
[3] A. Bertrand-Mathis , Comment écrire les nombres entiers dans une base qui n’est pas entière . Acta Math. Acad. Sci. Hungar. 54 ( 1989 ) 237 - 241 . Zbl 0695.10005 · Zbl 0695.10005 · doi:10.1007/BF01952053
[4] A. Bès , An extension of the Cobham-Semënov Theorem . J. Symb. Logic 65 ( 2000 ) 201 - 211 . Article | Zbl 0958.03025 · Zbl 0958.03025 · doi:10.2307/2586532 · http://minidml.mathdoc.fr/cgi-bin/location?id=00285794
[5] J.R. Büchi , Weak second-order arithmetic and finite automata . Z. Math. Logik Grundlagen Math. 6 ( 1960 ) 66 - 92 . MR 125010 | Zbl 0103.24705 · Zbl 0103.24705 · doi:10.1002/malq.19600060105
[6] V. Bruyère and G. Hansel , Bertrand numeration systems and recognizability . Theoret. Comput. Sci. 181 ( 1997 ) 17 - 43 . MR 1463527 | Zbl 0957.11015 · Zbl 0957.11015 · doi:10.1016/S0304-3975(96)00260-5
[7] A. Cobham , On the base-dependence of sets of numbers recognizable by finite automata . Math. Systems Theory 3 ( 1969 ) 186 - 192 . MR 250789 | Zbl 0179.02501 · Zbl 0179.02501 · doi:10.1007/BF01746527
[8] F. Durand , A generalization of Cobham’s Theorem . Theory Comput. Systems 31 ( 1998 ) 169 - 185 . Zbl 0895.68081 · Zbl 0895.68081 · doi:10.1007/s002240000084
[9] S. Eilenberg , Automata , Languages and Machines, Vol. A. Academic Press ( 1974 ). MR 530382 | Zbl 0317.94045 · Zbl 0317.94045
[10] S. Fabre , Une généralisation du théorème de Cobham . Acta Arithm. 67 ( 1994 ) 197 - 208 . Article | MR 1292734 | Zbl 0814.11015 · Zbl 0814.11015 · eudml:206626
[11] A.S. Fraenkel , Systems of numeration . Amer. Math. Monthly 92 ( 1985 ) 105 - 114 . MR 777556 | Zbl 0568.10005 · Zbl 0568.10005 · doi:10.2307/2322638
[12] Ch. Frougny , Representation of numbers and finite automata . Math. Systems Theory 25 ( 1992 ) 37 - 60 . MR 1139094 | Zbl 0776.11005 · Zbl 0776.11005 · doi:10.1007/BF01368783
[13] Ch. Frougny , Conversion between two multiplicatively dependent linear numeration systems , in Proc. of LATIN 02. Springer-Verlag, Lectures Notes in Comput. Sci. 2286 ( 2002 ) 64 - 75 . MR 1966116 | Zbl pre02086219 · Zbl 1152.11303 · http://link.springer.de/link/service/series/0558/bibs/2286/22860064.htm
[14] Ch. Frougny and J. Sakarovitch , Automatic conversion from Fibonacci representation to representation in base $\varphi $, and a generalization . Internat. J. Algebra Comput. 9 ( 1999 ) 351 - 384 . MR 1723473 | Zbl 1040.68061 · Zbl 1040.68061 · doi:10.1142/S0218196799000230
[15] Ch. Frougny and B. Solomyak , On Representation of Integers in Linear Numeration Systems , in Ergodic theory of ${\mbox {\boldmath {\bf Z}}}^d$-Actions, edited by M. Pollicott and K. Schmidt. Cambridge University Press, London Math. Soc. Lecture Note Ser. 228 ( 1996 ) 345 - 368 . MR 1411227 | Zbl 0856.11007 · Zbl 0856.11007
[16] Ch. Frougny and B. Solomyak , On the context-freeness of the $\theta $-expansions of the integers . Internat. J. Algebra Comput. 9 ( 1999 ) 347 - 350 . MR 1723472 | Zbl 1041.11008 · Zbl 1041.11008 · doi:10.1142/S0218196799000229
[17] G. Hansel , Systèmes de numération indépendants et syndéticité . Theoret. Comput. Sci. 204 ( 1998 ) 119 - 130 . MR 1637516 | Zbl 0952.68073 · Zbl 0952.68073 · doi:10.1016/S0304-3975(98)00035-8
[18] M. Hollander , Greedy numeration systems and regularity . Theory Comput. Systems 31 ( 1998 ) 111 - 133 . MR 1491655 | Zbl 0895.68088 · Zbl 0895.68088 · doi:10.1007/s002240000082
[19] D. Lind and B. Marcus , An Introduction to Symbolic Dynamics . Cambridge University Press ( 1995 ). MR 1369092 | Zbl 1106.37301 · Zbl 1106.37301 · doi:10.1017/CBO9780511626302
[20] M. Lothaire , Algebraic Combinatorics on Words . Cambridge University Press ( 2002 ). MR 1905123 | Zbl 1001.68093 · Zbl 1001.68093
[21] W. Parry , On the $\beta $-expansions of real numbers . Acta Math. Acad. Sci. Hungar. 11 ( 1960 ) 401 - 416 . MR 142719 | Zbl 0099.28103 · Zbl 0099.28103 · doi:10.1007/BF02020954
[22] Y. Puri and T. Ward , A dynamical property unique to the Lucas sequence . Fibonacci Quartely 39 ( 2001 ) 398 - 402 . MR 1866354 | Zbl 1018.37009 · Zbl 1018.37009
[23] Y. Puri and T. Ward , Arithmetic and growth of periodic orbits . J. Integer Sequences 4 ( 2001 ), Article 01.2.1. MR 1873399 | Zbl 1004.11013 · Zbl 1004.11013 · emis:journals/JIS/VOL4/WARD/short.html · eudml:49705
[24] A. Rényi , Representations for real numbers and their ergodic properties . Acta Math. Acad. Sci. Hungar. 8 ( 1957 ) 477 - 493 . MR 97374 | Zbl 0079.08901 · Zbl 0079.08901 · doi:10.1007/BF02020331
[25] A.L. Semënov , The Presburger nature of predicates that are regular in two number systems . Siberian Math. J. 18 ( 1977 ) 289 - 299 . MR 450050 | Zbl 0411.03054 · Zbl 0411.03054 · doi:10.1007/BF00967164
[26] J. Shallit , Numeration systems, linear recurrences, and regular sets . Inform. Comput. 113 ( 1994 ) 331 - 347 . MR 1285236 | Zbl 0810.11006 · Zbl 0810.11006 · doi:10.1006/inco.1994.1076