Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-11T08:58:32.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic point equivalence and Pisot numbers

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Low-dimensional dynamical systems

Published online by Cambridge University Press:  11 July 2019

SHIGEKI AKIYAMA Open the ORCID record for SHIGEKI AKIYAMA [Opens in a new window] ,
HAJIME KANEKO  and
DONG HAN KIM Open the ORCID record for DONG HAN KIM [Opens in a new window]
SHIGEKI AKIYAMA
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki305-8571, Japan email akiyama@math.tsukuba.ac.jp, kanekoha@math.tsukuba.ac.jp
HAJIME KANEKO
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki305-8571, Japan email akiyama@math.tsukuba.ac.jp, kanekoha@math.tsukuba.ac.jp
DONG HAN KIM
Affiliation:
Department of Mathematics Education, Dongguk University – Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul04620, Korea email kim2010@dongguk.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]{x1D6FD}x\}$ on $[0,1]$ and let $S:[0,1]\rightarrow [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$. We give a sufficient condition for $T$ and $S$ to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.

Keywords

normal numberbeta expansiongeneric pointPisot number

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3169 - 3180
Check for updates
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Airey, D., Jackson, S., Kwietniak, D. and Mance, B.. Borel complexity of sets of normal numbers via generic points in subshifts with specification. Preprint, 2018, arXiv:1811.04450v1.Google Scholar
Bertrand-Mathis, A.. Développement en base 𝜃; répartition modulo un de la suite (x𝜃n)n≥0 ; langages codés et 𝜃-shift. Bull. Soc. Math. France 114 (1986), 271–323.Google Scholar
Garsia, A. M.. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409–432.Google Scholar
Góra, P.. Invariant densities for generalized 𝛽-maps. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1583–1598.Google Scholar
Góra, P.. Invariant densities for piecewise linear maps of the unit interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583.Google Scholar
Handelman, D.. Spectral radii of primitive integral companion matrices and log concave polynomials. Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135). American Mathematical Society, Providence, RI, 1992, pp. 231–237.Google Scholar
Ito, S. and Takahashi, Y.. Markov subshifts and realization of 𝛽-expansions. J. Math. Soc. Japan 26(1) (1974), 33–55.Google Scholar
Jäger, H.. On decimal expansions. Zahlentheorie (Tagung), Math. Forschungsinst. Oberwolfach, 1970 (Bereich Math. Forschungsinst., Oberwolfach, Heft 5). Bibliographisches Institut, Mannheim, 1971, pp. 67–75.Google Scholar
Jung, S. and Volkmann, B.. Remarks on a paper of Wagner. J. Number Theory 56(2) (1996), 329–335.Google Scholar
Kano, H. and Shiokawa, I.. Rings of normal and nonnormal numbers. Israel J. Math. 84(3) (1993), 403–416.Google Scholar
Ki, H. and Linton, T.. Normal numbers and subset of N with given densities. Fund. Math. 144(2) (1994), 163–179.Google Scholar
Kopf, C.. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput. 39(2) (1990), 123–144, part II.Google Scholar
Kowalski, Z.. Invariant measure for piecewise monotonic transformation has a positive lower bound on its support. Bull. Acad. Polon. Sci. Ser. Sci. Math. 27(1) (1979), 53–57.Google Scholar
Kraaikamp, C. and Nakada, H.. On a problem of Schweiger concerning normal numbers. J. Number Theory 86 (2001), 330–340.Google Scholar
Li, T. Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.Google Scholar
Maxfield, J. E.. Normal k-tuples. Pacific J. Math. 3 (1953), 189–196.Google Scholar
Moshchevitin, N. G. and Shkredov, I. D.. On the Pyatetskii–Shapiro criterion of normality. Math. Notes 73 (2003), 539–550.Google Scholar
Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
Parry, W.. Representations for real numbers. Acta Math. Acad. Sci. Hungar. 15 (1964), 95–105.Google Scholar
Pollington, A. D.. The Hausdorff dimension of a set of normal numbers. Pacific J. Math. 95 (1981), 193–204.Google Scholar
Postnikov, A. G.. Ergodic problems in the theory of congruences and of diophantine approximations. Trudy Mat. Inst. Steklov 82 (1966), 3–112 (in Russian); Engl. trans. Proc. Steklov Inst. Math. 82 (1966), 1–128.Google Scholar
Schmidt, W. M.. On normal numbers. Pacific J. Math. 10 (1960), 661–672.Google Scholar
Schweiger, F.. Normalität bezüglich zahlentheoretischer Transformationen. J. Number Theory 1 (1969), 390–397.Google Scholar
Sharkovsky, A. N. and Sivak, A. G.. Basin of attractors of trajectories. J. Difference Equ. Appl. 22(2) (2016), 159–163.Google Scholar
Vandehey, J.. On the joint normality of certain digit expansions. Preprint, 2014, arXiv:1408.0435.Google Scholar
Wagner, G.. On rings of numbers which are normal to one base but non-normal to another. J. Number Theory 54(2) (1995), 211–231.Google Scholar
Wall, D. D.. Normal numbers. PhD Thesis, University of California, Berkeley, 1949.Google Scholar