Normal Numbers and Symbolic Dynamics

  • Manfred Madritsch
Part of the Trends in Mathematics book series (TM)


The present chapter takes a dynamical point of view. The orbit of an element plays a central role in dynamics, and we can deduce several properties such as periodicity, uniqueness, randomness, etc. from the orbit. Starting with a description of the link between dynamical systems and numeration systems, we present the concept of normal and non-normal numbers providing different views on the dynamics of the system. Normal numbers are “normal” with respect to randomly chosen objects, whereas non-normal numbers and extreme variants thereof are examples of general objects from a topological point of view. In the following sections, we present how to obtain maximal randomness as well as constructing numbers with a given degree of chaos. Then we turn our attention to non-normal numbers. Since they are not completely random, we have to find a different measurement for analyzing their structure. The Hausdorff dimension will provide us with an interesting parameter in this context.


  1. 10.
    Albeverio, S., Pratsiovytyi, M., Torbin, G.: Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits. Ukraïn. Mat. Zh. 57(9), 1163–1170 (2005)MathSciNetzbMATHGoogle Scholar
  2. 30.
    Baek, I.S., Olsen, L.: Baire category and extremely non-normal points of invariant sets of IFS’s. Discrete Contin. Dyn. Syst. 27(3), 935–943 (2010)MathSciNetCrossRefGoogle Scholar
  3. 36.
    Barat, G., Berthé, V., Liardet, P., Thuswaldner, J.: Dynamical directions in numeration. Ann. Inst. Fourier (Grenoble) 56(7), 1987–2092 (2006)Google Scholar
  4. 80.
    Bertrand-Mathis, A.: Points génériques de Champernowne sur certains systèmes codes; application aux θ-shifts. Ergodic Theory Dyn. Syst. 8(1), 35–51 (1988)MathSciNetCrossRefGoogle Scholar
  5. 81.
    Bertrand-Mathis, A., Volkmann, B.: On (𝜖, k)-normal words in connecting dynamical systems. Monatsh. Math. 107(4), 267–279 (1989)MathSciNetCrossRefGoogle Scholar
  6. 82.
    Besicovitch, A.S.: The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers. Math. Z. 39, 146–156 (1934)MathSciNetCrossRefGoogle Scholar
  7. 85.
    Billingsley, P.: Hausdorff dimension in probability theory. Ill. J. Math. 4, 187–209 (1960)MathSciNetzbMATHGoogle Scholar
  8. 86.
    Billingsley, P.: Hausdorff dimension in probability theory. II. Ill. J. Math. 5, 291–298 (1961)MathSciNetGoogle Scholar
  9. 87.
    Billingsley, P.: Ergodic Theory and Information. Wiley, New York (1965)zbMATHGoogle Scholar
  10. 88.
    Blanchard, F., Hansel, G.: Systèmes codés. Theor. Comput. Sci. 44(1), 17–49 (1986)CrossRefGoogle Scholar
  11. 99.
    Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti Circ. Mat. Palermo 27, 247–271 (1909)CrossRefGoogle Scholar
  12. 103.
    Bowen, R.: Periodic points and measures for Axiom A diffeomorphisms. Trans. Am. Math. Soc. 154, 377–397 (1971)MathSciNetzbMATHGoogle Scholar
  13. 141.
    Champernowne, D.G.: The construction of decimals normal in the scale of ten. J. Lond. Math. Soc. s1–8, 254–260 (1933)Google Scholar
  14. 166.
    Copeland, A.H., Erdős, P.: Note on normal numbers. Bull. Am. Math. Soc. 52, 857–860 (1946)Google Scholar
  15. 167.
    Cornfeld, I.P., Fomin, S.V., Sinaı̆, Y.G.: Ergodic theory. Springer, New York (1982). Translated from the Russian by A.B. Sosinskiı̆Google Scholar
  16. 176.
    Dajani, K., Kraaikamp, C.: Ergodic Theory of Numbers. Carus Mathematical Monographs, vol. 29. Mathematical Association of America, Washington, DC (2002)Google Scholar
  17. 180.
    Davenport, H., Erdős, P.: Note on normal decimals. Can. J. Math. 4, 58–63 (1952)Google Scholar
  18. 193.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol. 527. Springer, Berlin (1976)Google Scholar
  19. 210.
    Eggleston, H.G.: The fractional dimension of a set defined by decimal properties. Q. J. Math. Oxford Ser. 20, 31–36 (1949)MathSciNetCrossRefGoogle Scholar
  20. 259.
    Goodwyn, L.W.: Topological entropy bounds measure-theoretic entropy. Proc. Am. Math. Soc. 23, 679–688 (1969)MathSciNetCrossRefGoogle Scholar
  21. 307.
    Hyde, J., Laschos, V., Olsen, L., Petrykiewicz, I., Shaw, A.: Iterated Cesàro averages, frequencies of digits, and Baire category. Acta Arith. 144(3), 287–293 (2010)MathSciNetCrossRefGoogle Scholar
  22. 361.
    Kraaikamp, C., Nakada, H.: On normal numbers for continued fractions. Ergodic Theory Dyn. Syst. 20(5), 1405–1421 (2000)MathSciNetCrossRefGoogle Scholar
  23. 377.
    Levy, P.: Sur les lois de probabilité dont dependent les quotients complets et incomplets d’une fraction continue. Bull. Soc. Math. France 57, 178–194 (1929)MathSciNetCrossRefGoogle Scholar
  24. 378.
    Lewis, J.T., Pfister, C.E., Russell, R.P., Sullivan, W.G.: Reconstruction sequences and equipartition measures: an examination of the asymptotic equipartition property. IEEE Trans. Inform. Theory 43(6), 1935–1947 (1997)MathSciNetCrossRefGoogle Scholar
  25. 379.
    Lewis, J.T., Pfister, C.E., Sullivan, W.G.: Large deviations and the thermodynamic formalism: a new proof of the equivalence of ensembles. In: Fannes, M., Maes, C., Verbeure, A. (eds.) On Three Levels: Micro-, Meso-, and Macro-Approaches in Physics, pp. 183–192. Springer US, Boston, MA (1994)CrossRefGoogle Scholar
  26. 380.
    Liao, L., Ma, J., Wang, B.: Dimension of some non-normal continued fraction sets. Math. Proc. Camb. Philos. Soc. 145(1), 215–225 (2008)MathSciNetCrossRefGoogle Scholar
  27. 394.
    Lüroth, J.: Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Ann. 21(3), 411–423 (1883)MathSciNetCrossRefGoogle Scholar
  28. 398.
    Madritsch, M.G.: Non-normal numbers with respect to Markov partitions. Discret. Contin. Dyn. Syst. 34(2), 663–676 (2014)MathSciNetCrossRefGoogle Scholar
  29. 399.
    Madritsch, M.G., Mance, B.: Construction of μ-normal sequences. Monatsh. Math. 179(2), 259–280 (2016)MathSciNetCrossRefGoogle Scholar
  30. 400.
    Madritsch, M.G., Petrykiewicz, I.: Non-normal numbers in dynamical systems fulfilling the specification property. Discret. Contin. Dyn. Syst. 34(11), 4751–4764 (2014)MathSciNetCrossRefGoogle Scholar
  31. 401.
    Madritsch, M.G., Thuswaldner, J.M., Tichy, R.F.: Normality of numbers generated by the values of entire functions. J. Number Theory 128(5), 1127–1145 (2008)MathSciNetCrossRefGoogle Scholar
  32. 434.
    Nakada, H.: Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 4(2), 399–426 (1981)MathSciNetCrossRefGoogle Scholar
  33. 435.
    Nakai, Y., Shiokawa, I.: A class of normal numbers. Jpn. J. Math. (N.S.) 16(1), 17–29 (1990)Google Scholar
  34. 436.
    Nakai, Y., Shiokawa, I.: Discrepancy estimates for a class of normal numbers. Acta Arith. 62(3), 271–284 (1992)MathSciNetCrossRefGoogle Scholar
  35. 437.
    Nakai, Y., Shiokawa, I.: Normality of numbers generated by the values of polynomials at primes. Acta Arith. 81, 345–356 (1997)MathSciNetCrossRefGoogle Scholar
  36. 457.
    Olsen, L.: Extremely non-normal continued fractions. Acta Arith. 108(2), 191–202 (2003)MathSciNetCrossRefGoogle Scholar
  37. 458.
    Olsen, L.: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. (9) 82(12), 1591–1649 (2003)Google Scholar
  38. 459.
    Olsen, L.: Extremely non-normal numbers. Math. Proc. Camb. Philos. Soc. 137(1), 43–53 (2004)MathSciNetCrossRefGoogle Scholar
  39. 460.
    Olsen, L., Winter, S.: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II. Non-linearity, divergence points and Banach space valued spectra. Bull. Sci. Math. 131(6), 518–558 (2007)Google Scholar
  40. 469.
    Parry, W.: On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11, 401–416 (1960)MathSciNetCrossRefGoogle Scholar
  41. 472.
    Pavlov, R.: On intrinsic ergodicity and weakenings of the specification property. Adv. Math. 295, 250–270 (2016)MathSciNetCrossRefGoogle Scholar
  42. 478.
    Pfister, C.E., Sullivan, W.G.: Billingsley dimension on shift spaces. Nonlinearity 16(2), 661–682 (2003)MathSciNetCrossRefGoogle Scholar
  43. 479.
    Pfister, C.E., Sullivan, W.G.: On the topological entropy of saturated sets. Ergodic Theory Dyn. Syst. 27(3), 929–956 (2007)MathSciNetCrossRefGoogle Scholar
  44. 500.
    Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477–493 (1957)MathSciNetCrossRefGoogle Scholar
  45. 510.
    Rosen, D.: A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21, 549–563 (1954)MathSciNetCrossRefGoogle Scholar
  46. 526.
    Scheerer, A.M.: Normality in Pisot numeration systems. Ergodic Theory Dyn. Syst. 37(2), 664–672 (2017)MathSciNetCrossRefGoogle Scholar
  47. 527.
    Schiffer, J.: Discrepancy of normal numbers. Acta Arith. 47(2), 175–186 (1986)MathSciNetCrossRefGoogle Scholar
  48. 533.
    Schweiger, F.: Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications/The Clarendon Press/Oxford University Press, New York (1995)zbMATHGoogle Scholar
  49. 572.
    Vaaler, J.D.: Some extremal functions in Fourier analysis. Bull. Am. Math. Soc. (N.S.) 12(2), 183–216 (1985)Google Scholar
  50. 573.
    Vandehey, J.: A simpler normal number construction for simple Lüroth series. J. Integer Seq. 17(6), Article 14.6.1, 18 (2014)Google Scholar
  51. 575.
    Volkmann, B.: Über Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind. VI. Math. Z. 68, 439–449 (1958)CrossRefGoogle Scholar
  52. 579.
    Walters, P.: An Introduction to Ergodic Theory. Springer, New York (1982)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Élie Cartan de LorraineUniverstité de LorraineVandœuvre-Lès-Nancy CedexFrance

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