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

Title & Authors
THE ORBIT OF A β-TRANSFORMATION CANNOT LIE IN A SMALL INTERVAL
Kwon, Do-Yong;

Abstract
For $\small{{\beta}}$ > 1, let $\small{T_{\beta}}$ : [0, 1] $\small{{\rightarrow}}$ [0, 1) be the $\small{{\beta}}$-transformation. We consider an invariant $\small{T_{\beta}}$-orbit closure contained in a closed interval with diameter 1/$\small{{\beta}}$, then define a function $\small{{\Xi}({\alpha},{\beta})}$ by the supremum such $\small{T_{\beta}}$-orbit with frequency $\small{{\alpha}}$ in base $\small{{\beta}}$, i.e., the maximum value in $\small{T_{\beta}}$-orbit closure. This paper effectively determines the maximal domain of $\small{{\Xi}}$, and explicitly specifies all possible minimal intervals containing $\small{T_{\beta}}$-orbits.
Keywords
$\small{{\beta}}$-expansion;$\small{{\beta}}$-transformation;Sturmian word;Christoffel word;
Language
English
Cited by
1.
Moments of discrete measures with dense jumps induced by β -expansions, Journal of Mathematical Analysis and Applications, 2013, 399, 1, 1
2.
A two-dimensional singular function via Sturmian words in base β, Journal of Number Theory, 2013, 133, 11, 3982
3.
A one-parameter family of Dirichlet series whose coefficients are Sturmian words, Journal of Number Theory, 2015, 147, 824
4.
Exceptional parameters of linear mod one transformations and fractional parts {ξ(p/q)n}, Comptes Rendus Mathematique, 2015, 353, 4, 291
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.

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.

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.

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.

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.

8.
E. M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138-153.

9.
D. P. Chi and D. Y. Kwon, Sturmian words, ${\beta}$-shifts, and transcendence, Theoret. Comput. Sci. 321 (2004), no. 2-3, 395-404.

10.
S. Ferenczi and C. Mauduit, Transcendence of numbers with a low complexity expansion, J. Number Theory 67 (1997), no. 2, 146-161.

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.

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.

17.
M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42.

18.
W. Parry, On the ${\beta}$-expansion of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.

19.
A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493.