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METRIC THEOREM AND HAUSDORFF DIMENSION ON RECURRENCE RATE OF LAURENT SERIES

  • Hu, Xue-Hai ;
  • Li, Bing ;
  • Xu, Jian
  • Received : 2013.01.16
  • Published : 2014.01.31

Abstract

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let $E_{{\alpha},{\beta}}$ be the set of points with lower and upper recurrence rates ${\alpha},{\beta}$, ($0{\leq}{\alpha}{\leq}{\beta}{\leq}{\infty}$), we prove that all the sets $E_{{\alpha},{\beta}}$, are of full Hausdorff dimension. Then the recurrence sets $E_{{\alpha},{\beta}}$ have constant multifractal spectra.

Keywords

recurrence rate;pointwise dimension;continued fractions;Laurent series;Hausdorff dimension

References

  1. E. Artin, Quadratische Korper im Gebiete der hoheren Kongruenzen, I-II, Math. Z. 19 (1924), no. 1, 153-246. https://doi.org/10.1007/BF01181074
  2. L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincare recurrence, Comm. Math. Phys. 219 (2001), no. 2, 443-463. https://doi.org/10.1007/s002200100427
  3. L. Barreira and B. Saussol, Product structure of Poincare recurrence, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 33-61.
  4. V. Berthe and H. Nakada, On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Expo. Math. 18 (2000), no. 4, 257-284.
  5. M. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 (1993), no. 3, 617-631. https://doi.org/10.1007/BF01244320
  6. K. J. Falconer, Fractal Geometry, Mathematical Foundations and Application, Wiley, 1990.
  7. D. J. Feng and J. Wu, The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity 14 (2001), no. 1, 81-85. https://doi.org/10.1088/0951-7715/14/1/304
  8. X. H. Hu, B.W. Wang, J. Wu, and Y. L. Yu, Cantor sets determined by partial quotients of continued fractions of Laurent series, Finite Fields Appl. 14 (2008), no. 2, 417-437. https://doi.org/10.1016/j.ffa.2007.04.002
  9. K. S. Lau and L. Shu, The spectrum of Poincare recurrence, Ergodic Theory Dynam. Systems 28 (2008), no. 6, 1917-1943. https://doi.org/10.1017/S0143385707001095
  10. H. Niederreiter, The probabilistic theory of linear complexity, Advances in cryptology EUROCRYPT '88 (Davos, 1988), 191209, Lecture Notes in Comput. Sci., 330, Springer, Berlin, 1988.
  11. H. Niederreiter and M. Vielhaber, Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles, J. Complexity. 13 (1997), no. 3, 353-383. https://doi.org/10.1006/jcom.1997.0451
  12. D. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory 39 (1993), no. 1 78-83. https://doi.org/10.1109/18.179344
  13. R. Paysant-Leroux and E. Dubois, Etude metrique de l'algorithme de Jacobi-Perron dans un corps de series formelles, (French) C. R. Acad. Sci. Paris Ser. A-B 275 (1972), A683-A686.
  14. B. Saussol and J. Wu, Recurrence spectrum in smooth dynamical system, Nonlinearity 16 (2003), no. 6, 1991-2001. https://doi.org/10.1088/0951-7715/16/6/306
  15. W. M. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith. 95 (2000), no. 2, 139-166. https://doi.org/10.4064/aa-95-2-139-166
  16. J. Wu, Hausdorff dimensions of bounded type continued fraction sets of Laurent series, Finite Fields Appl. 13 (2007), no. 1, 20-30. https://doi.org/10.1016/j.ffa.2005.05.003