# THE ORBIT OF A β-TRANSFORMATION CANNOT LIE IN A SMALL INTERVAL

• Kwon, Do-Yong
• Received : 2011.07.23
• Published : 2012.07.01
• 55 7

#### Abstract

For ${\beta}$ > 1, let $T_{\beta}$ : [0, 1] ${\rightarrow}$ [0, 1) be the ${\beta}$-transformation. We consider an invariant $T_{\beta}$-orbit closure contained in a closed interval with diameter 1/${\beta}$, then define a function ${\Xi}({\alpha},{\beta})$ by the supremum such $T_{\beta}$-orbit with frequency ${\alpha}$ in base ${\beta}$, i.e., the maximum value in $T_{\beta}$-orbit closure. This paper effectively determines the maximal domain of ${\Xi}$, and explicitly specifies all possible minimal intervals containing $T_{\beta}$-orbits.

#### Keywords

${\beta}$-expansion;${\beta}$-transformation;Sturmian word;Christoffel word

#### References

1. B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers. I. Expansions in integer bases, Ann. of Math. (2) 165 (2007), no. 2, 547-565. https://doi.org/10.4007/annals.2007.165.547
2. J.-P. Allouche and A. Glen, Distribution modulo 1 and the lexicographic world, Ann. Sci. Math. Quebec 33 (2009), no. 2, 125-143.
3. J.-P. Allouche and A. Glen, Extremal properties of (epi)Sturmian sequences and distribution modulo 1, Enseign. Math. (2) 56 (2010), no. 3-4, 365-401. https://doi.org/10.4171/LEM/56-3-5
4. J. Berstel, A. Lauve, C. Reutenauer, and F. V. Saliola, Combinatorics on Words, American Mathematical Society, 2009.
5. F. Blanchard, ${\beta}$-expansions and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), no. 2, 131-141. https://doi.org/10.1016/0304-3975(89)90038-8
6. Y. Bugeaud and A. Dubickas, Fractional parts of powers and Sturmian words, C. R. Math. Acad. Sci. Paris 341 (2005), no. 2, 69-74. https://doi.org/10.1016/j.crma.2005.06.007
7. S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil's staircase, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 451-481. https://doi.org/10.1017/S0305004100072236
8. E. M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138-153. https://doi.org/10.1007/BF01762232
9. D. P. Chi and D. Y. Kwon, Sturmian words, ${\beta}$-shifts, and transcendence, Theoret. Comput. Sci. 321 (2004), no. 2-3, 395-404. https://doi.org/10.1016/j.tcs.2004.03.035
10. S. Ferenczi and C. Mauduit, Transcendence of numbers with a low complexity expansion, J. Number Theory 67 (1997), no. 2, 146-161. https://doi.org/10.1006/jnth.1997.2175
11. L. Flatto, Z-numbers and ${\beta}$-transformations, Symbolic dynamics and its applications (New Haven, CT, 1991), 181-201, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.
12. L. Flatto, J. C. Lagarias, and A. D. Pollington, On the range of fractional parts {${\xi}(p/q)^n$}, Acta Arith. 70 (1995), no. 2, 125-147. https://doi.org/10.4064/aa-70-2-125-147
13. D. Y. Kwon, A devil's staircase from rotations and irrationality measures for Liouville numbers, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 739-756.
14. D. Y. Kwon, A two dimensional singular function via Sturmian words in base ${\beta}$, preprint (2011). Available at http://www.math.jnu.ac.kr/doyong/paper/twosing_Apr2011.pdf
15. M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002.
16. K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (1968), 313-321. https://doi.org/10.1017/S1446788700005371
17. M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42. https://doi.org/10.2307/2371431
18. W. Parry, On the ${\beta}$-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. https://doi.org/10.1007/BF02020954
19. A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. https://doi.org/10.1007/BF02020331

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2. A two-dimensional singular function via Sturmian words in base β vol.133, pp.11, 2013, https://doi.org/10.1016/j.jnt.2012.11.008
3. A one-parameter family of Dirichlet series whose coefficients are Sturmian words vol.147, 2015, https://doi.org/10.1016/j.jnt.2014.08.018
4. Exceptional parameters of linear mod one transformations and fractional parts {ξ(p/q)n} vol.353, pp.4, 2015, https://doi.org/10.1016/j.crma.2015.01.017
5. Sturmian words and Cantor sets arising from unique expansions over ternary alphabets pp.1469-4417, 2018, https://doi.org/10.1017/etds.2017.136

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)