PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

doi:10.1016/S0252-9602(16)30113-8

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (1): 33-46.doi: 10.1016/S0252-9602(16)30113-8

• Articles • Previous Articles Next Articles

PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

Yuehua GE^1,2, Bo TAN¹

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
2. School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China

Received:2016-01-07 Revised:2016-05-05 Online:2017-02-25 Published:2017-02-25
Contact: Bo TAN,E-mail:tanbo@mail.hust.edu.cn E-mail:tanbo@mail.hust.edu.cn
About author:Yuehua GE,E-mail:geyuehua1001@126.com;
Supported by:
This work was supported by NSFC (11171123, 11222111).

Abstract

Abstract:

Let m≥1 be an integer, 1 < β < m+1. A sequence ε1ε2ε3… with ε_i∈{0, 1, …,m} is called a β-expansion of a real number x if x=∑_i(ε_i)/(βⁱ). It is known that when the base β is smaller than the generalized golden ration, any number has uncountably many expansions, while when β is larger, there are numbers which has unique expansion. In this paper, we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period. We prove that such bases form an open interval, moreover, any two such open intervals have inclusion relationship according to the Sharkovskiǐ ordering between the given minimal periods. We remark that our result answers an open question posed by Baker, and the proof for the case m=1 is due to Allouche, Clarke and Sidorov.

Key words: beta-expansions, periodic expansions, unique expansion, the Sharkovskii ordering

Yuehua GE, Bo TAN. PERIODICITY OF THE UNIVOQUE β-EXPANSIONS[J].Acta mathematica scientia,Series B, 2017, 37(1): 33-46.

Trendmd

References

[1] Renyi A. Representions for real numbers and their ergodic properties. Acta Math Acad Sci Hungar, 1957, 8:477-493
[2] Parry W. On the β-expansion of real number. Acta Math Acad Sci Hungar, 1960, 11:401-416
[3] Baker S. Generalized golden ratios over integer alphabets. Integers, 2014, 14(A15):28 pp
[4] Dajani K, Ramawadh S D. Symbolic dynamics of (-β)-expansions. J Integer Seq, 2012, 15(2)
[5] Ito S, Takahashi Y. Markov subshifts and realization of β-expansions. J Math Soc Japan, 1974, 26:33-55
[6] Komornik V, Loreti P. On the topological structure of univoque sets. J Number Theory, 2007, 122(7):157-183
[7] Nakano F, Sadahiro T. A (-β)-expansion associated to Sturmian sequences. Integers, 2012, 12(4):571-599
[8] Sidorov N. Almost every number has a continuum of β-expansion. Amer Math Monthly, 2003, 110(9):838-842
[9] Baatz M, Komornik V. Unique expansions in integer bases with extended alphabets. Publ Math, 2011, 79(3/4):251-267
[10] Komornik V. Expansions in noninteger bases. Integers, 2011, 11B(A9):30 pp
[11] Komornik V, Loreti P. Unique developments in non-integer bases. Amer Math Monthly, 1998, 105(7):636-639
[12] Komornik V, Loreti P. Universal expansions in negative and complex bases. Integers, 2015, 10(A50):669-679
[13] Kong D R, Li W X. Hausdorff dimension of unique beta expansions. Nonlinearity, 2015, 28(1):187-209
[14] Lü F, Tan B, Wu J. Univoque sets for real numbers. Fund Math, 2014, 227(1):69-83
[15] Erdas P, Horváth M, Jóo I. On the uniqueness of the expansion 1=q^-n_i. Acta Math Hungar, 1991, 58:333-342
[16] de Vries M, Komornik V. Unique expansions of real numbers. Adv Math, 2009, 221(2):390-427
[17] de Vries M. On the number of unique expansions in non-integer bases. Topology Appl, 2009, 156(3):652-657
[18] Erds P, Jóo I, Komornik V. Characterization of the unique expansions 1=q^-n_i and related problems. Bull Soc Math Fr, 1990, 118:377-390
[19] Daroczy Z, Katai I. Univoque sequences. Publ Math Debrecen, 1993, 42(3/4):397-407
[20] Sidorov N, Vershik A. Erdogic properties of the Erds measure, the entropy of the golden shift, and related problems. Monatsh Math, 2012, 126(3):717-751
[21] Ge Y H, Tan B. Numbers with countable expansions in base of generalized golden ratios. arXiv:1504.01704[math.], Apr 5, 2015
[22] Allouche J, Shallit J. The ubiquitous Prouhet-Thue-Morse sequence//Sequences and Their Applications (Singapore, 1998). Springer Ser Discrete Math Theor Comput Sci. London:Springer, 1999:1-16
[23] Allouche J -P, Cosnard M. The Komornik-Loreti constant is transcendental. Amer Math Monthly, 2000, 107(5):448-449
[24] Allouche J -P, Cosnard M. Non-integer bases, iteration of continuous functions, and an arithmetic selfsimilar set. Acta Math Hungar, 2001, 91(4):325-332
[25] Komornik V, Loreti P. Subexpansion, supexpansions and uniqueness prperties in non-integer bases. Period Math Hung, 2002, 44:197-218
[26] Allouche J, Clarke M, Sidorov N. Periodic unique beta-expansions:the Sharkovski? ordering. Ergod Theory Dynam Sys, 2009, 29(4):1055-1074
[27] Kong D R, Li W X, Dekking F M. Interesections of homogeneous Cantor sets and beta-expansions. Nonlinearity, 2010, 23(11):2815-2834
[28] Queffélec M. Substitution Dynamical Systems-Spectral Analysis. Lecture Notes in Mathematics 1294. 2nd ed. Berlin:Springer-Verlag, 2010

PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 0

Metrics

Comments

Recommended 10