Skip to main content Accessibility help

Generic point equivalence and Pisot numbers



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.



Hide All
[1] 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.
[2] Bertrand-Mathis, A.. Développement en base 𝜃; répartition modulo un de la suite (x𝜃 n )n≥0 ; langages codés et 𝜃-shift. Bull. Soc. Math. France 114 (1986), 271323.
[3] Garsia, A. M.. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409432.
[4] Góra, P.. Invariant densities for generalized 𝛽-maps. Ergod. Th. & Dynam. Sys. 27(5) (2007), 15831598.
[5] Góra, P.. Invariant densities for piecewise linear maps of the unit interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 15491583.
[6] 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. 231237.
[7] Ito, S. and Takahashi, Y.. Markov subshifts and realization of 𝛽-expansions. J. Math. Soc. Japan 26(1) (1974), 3355.
[8] Jäger, H.. On decimal expansions. Zahlentheorie (Tagung), Math. Forschungsinst. Oberwolfach, 1970 (Bereich Math. Forschungsinst., Oberwolfach, Heft 5) . Bibliographisches Institut, Mannheim, 1971, pp. 6775.
[9] Jung, S. and Volkmann, B.. Remarks on a paper of Wagner. J. Number Theory 56(2) (1996), 329335.
[10] Kano, H. and Shiokawa, I.. Rings of normal and nonnormal numbers. Israel J. Math. 84(3) (1993), 403416.
[11] Ki, H. and Linton, T.. Normal numbers and subset of N with given densities. Fund. Math. 144(2) (1994), 163179.
[12] Kopf, C.. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput. 39(2) (1990), 123144, part II.
[13] 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), 5357.
[14] Kraaikamp, C. and Nakada, H.. On a problem of Schweiger concerning normal numbers. J. Number Theory 86 (2001), 330340.
[15] Li, T. Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183192.
[16] Maxfield, J. E.. Normal k-tuples. Pacific J. Math. 3 (1953), 189196.
[17] Moshchevitin, N. G. and Shkredov, I. D.. On the Pyatetskii–Shapiro criterion of normality. Math. Notes 73 (2003), 539550.
[18] Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401416.
[19] Parry, W.. Representations for real numbers. Acta Math. Acad. Sci. Hungar. 15 (1964), 95105.
[20] Pollington, A. D.. The Hausdorff dimension of a set of normal numbers. Pacific J. Math. 95 (1981), 193204.
[21] Postnikov, A. G.. Ergodic problems in the theory of congruences and of diophantine approximations. Trudy Mat. Inst. Steklov 82 (1966), 3112 (in Russian); Engl. trans. Proc. Steklov Inst. Math. 82 (1966), 1–128.
[22] Schmidt, W. M.. On normal numbers. Pacific J. Math. 10 (1960), 661672.
[23] Schweiger, F.. Normalität bezüglich zahlentheoretischer Transformationen. J. Number Theory 1 (1969), 390397.
[24] Sharkovsky, A. N. and Sivak, A. G.. Basin of attractors of trajectories. J. Difference Equ. Appl. 22(2) (2016), 159163.
[25] Vandehey, J.. On the joint normality of certain digit expansions. Preprint, 2014, arXiv:1408.0435.
[26] Wagner, G.. On rings of numbers which are normal to one base but non-normal to another. J. Number Theory 54(2) (1995), 211231.
[27] Wall, D. D.. Normal numbers. PhD Thesis, University of California, Berkeley, 1949.


MSC classification

Generic point equivalence and Pisot numbers



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed