DOI QR코드

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NORMAL NUMBERS MOD 2 ON THE LOGISTIC MAP

Ahn, Young-Ho

  • Received : 2013.10.23
  • Accepted : 2013.11.11
  • Published : 2013.12.25

Abstract

We show that for the logistic map, almost every x is a normal number mod 2 with respect to all intervals except for $[a,b]=[\frac{1}{4},1]\;or\;[a,b]=[\frac{1}{2}-\frac{\sqrt{3}}{4},\frac{1}{2}+\frac{\sqrt{3}}{4}]$.

Keywords

normal number mod 2;logistic map;tent map;ergodic

References

  1. Y. Ahn and G. H. Choe, Spectral types of skewed Bernoulli shift, Proc. Amer. Math. Soc. 128 (2000), 503-510. https://doi.org/10.1090/S0002-9939-99-04990-4
  2. Y. Ahn, A class of compact group extensions of ${\beta}$-transformations, J. Math. Anal. Appl. 376 (2011), 154-161. https://doi.org/10.1016/j.jmaa.2010.08.076
  3. G. H. Choe, Computational Ergodic Theory, Springer-Verlag, 2005.
  4. G. H. Choe, T. Hamachi and H. Nakada Skew product and mod 2 normal numbers, Studia Math. 165 (2004), 53-60 https://doi.org/10.4064/sm165-1-4
  5. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1986.
  6. W. A. Veech, strict ergodicity of uniform distribution and Kronecker-Weyl theorem mod 2, Tran. Amer. Math. Soc. 140 (1969), 1-33.
  7. P.Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.