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Algorithmica

, Volume 29, Issue 1–2, pp 262–306 | Cite as

Dynamical sources in information theory: Fundamental intervals and word prefixes

  • B. Vallée
Article

Abstract

A quite general model of source that comes from dynamical systems theory is introduced. Within this model, some basic problems of algorithmic information theory contexts are analysed. The main tool is a new object, the generalized Ruelle operator, which can be viewed as a “generating” operator for fundamental intervals (associated to information sharing common prefixes). Its dominant spectral objects are linked with important parameters of the source, such as the entropy, and play a central rôle in all the results.

Key Words

Information theory Dynamical systems Transfer operator Sources Entropy Fundamental intervals 

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References

  1. [1]
    Avnaim, F., Boissonnat, J.-D., Devillers, O., Preparata, F., and Yvinec, M. Evaluation of a new method to compute signs of determinants. InProceedings of the Eleventh Annual ACM Symposium on Computational Geometry, 1995, pp. C16–C17. Full paper inAlgorithmica,17 (1997), 111–132.Google Scholar
  2. [2]
    Bedford, T., Keane, M., and Series, C., eds.Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Oxford University Press, 1991.Google Scholar
  3. [3]
    Beeler, M., Gosper, R. W., and Schroeppel, R. HAKMEM. Memorandum 239, Artificial Intelligence Laboratory, M.I.T., Feb. 1972.Google Scholar
  4. [4]
    Billingsley, P.Ergodic Theory and Information. Wiley, New York, 1965.zbMATHGoogle Scholar
  5. [5]
    Bogomolny, E. B., and Carioli, M. Quantum maps from transfer operators.Physica D 67 (1993), 88–112.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Daudé, H., Flajolet, P., and Vallée, B. An average-case analysis of the Gaussian algorithm for lattice reduction.Combinatorics, Probability and Computing 6 (1997), 397–433.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Delange, H. Généralisation du Théorème d’Ikehara.Annales Scientifiques de l’Ecole Normale Supériore 71 (1954), 213–242.zbMATHMathSciNetGoogle Scholar
  8. [8]
    Devroye, L. A probabilistic analysis of the height of tries and of the complexity of triesort.Acta Informatica,21 (1984), 229–237.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Efrat, I. Dynamics of the continued fraction map and the spectral theory ofSL(2,Z).Inventiones Mathematicae 114 (1993), 207–218.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Faivre, C. Distribution of Lévy’s constants for quadratic numbers.Acta Arithmetica 61(1) (1992), 13–34.zbMATHMathSciNetGoogle Scholar
  11. [11]
    Fayolle, G., Flajolet, P., and Hofri, M. On a functional equation arising in the analysis of a protocol for a multi-accessbroadcast channel.Advances in Applied Probability 18 (1986), 441–472.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Flajolet, P., Gourdon, X., and Dumas, P. Mellin transforms and asymptotics: harmonic sums.Theoretical Computer Science 144(1–2) (1995), 3–58.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Flajolet, P., and Vallée, B. Continued fraction algorithms, functional operators and structure constants.Theoretical Computer Science 194 (1998), 1–34.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Grothendieck, A. Produits tensoriels topologiques et espaces nucléaires.Memoirs of the American Mathematical Society 16 (1955).Google Scholar
  15. [15]
    Grothendieck, A. La théorie de Fredholm.Bulletin de la Société Mathématique de France 84 (1956), 319–384.zbMATHMathSciNetGoogle Scholar
  16. [16]
    Hwang, H.-K. Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques. Ph.D. thesis, École Polytechnique, Dec. 1994.Google Scholar
  17. [17]
    Kato, T.Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1980.zbMATHGoogle Scholar
  18. [18]
    Knuth, D.E.Computers and Typesetting, MF: the program, Volume D. Addison-Wesley, Reading, MA, 1986.Google Scholar
  19. [19]
    Krasnoselskii, M.Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964.Google Scholar
  20. [20]
    Lorch, E. R.Spectral Theory. Oxford University Press, New York, 1962.zbMATHGoogle Scholar
  21. [21]
    Mayer, D. H. On a ζ function related to the continued fraction transformation.Bulletin de la Société Mathématique de France 104 (1976), 195–203.zbMATHGoogle Scholar
  22. [22]
    Mayer, D. H. Spectral properties of certain composition operators arising in statistical mechanics,Communications in Mathematical Physics 68 (1979), 1–8.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Mayer, D. H. Continued fractions and related transformations. InErgodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane, and C. Series, eds. Oxford University Press, Oxford, 1991, pp. 175–222.Google Scholar
  24. [24]
    Mayer, D. H. Private communication.Google Scholar
  25. [25]
    Mischyavichyus, G. A. Estimate of the remainder in the limit theorem for the denominators of continued fractions. Litovskiǐ Matematičeskiǐ Sbornik21(3), (1987), 63–74.Google Scholar
  26. [26]
    Morita, T. Local limit theorem and distribution of periodic orbits of Lasota-Yorke transformations with infinite Markov partitions.Journal of the Mathematical Society of Japan 46(2) (1994), 309–343. Corrections, op. cit.47(1) (1997), 191–192.zbMATHMathSciNetGoogle Scholar
  27. [27]
    Philipp, W. Ein zentraler Grenzwertsatz mit Anwendungen auf die Zahlentheorie.Zeitschrift für Wahrscheinlichkeitstheorie 8 (1967), 195–203.Google Scholar
  28. [28]
    Pollicott, M. A complex Ruelle-Perron-Frobenius Theorem and two counterexamples.Ergodic Theory and Dynamical Systems 4 (1984), 135–146.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    Ruelle, D.Thermodynamic Formalism. Addison-Wesley, Reading, MA, 1978.zbMATHGoogle Scholar
  30. [30]
    Ruelle, D.Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval. Volume 4 of CRM Monograph Series. American Mathematical Society, Providence, RI, 1994.zbMATHGoogle Scholar
  31. [31]
    Schwartz, H. Composition operators inH p. Ph.D. Thesis, University of Toledo.Google Scholar
  32. [32]
    Shannon, C. A mathematical theory of communication.Bell Systems Technical Journal 27 (1948), 379–423, 623–656.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Shapiro, J.Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.Google Scholar
  34. [34]
    Shapiro, J. Compact composition operators on spaces of boundary regular holomorphic functions.Proceedings of the AMS 100 (1997), 49–57.CrossRefGoogle Scholar
  35. [35]
    Shapiro, J., and Taylor, P.D. Compact, nuclear, and Hilbert-Schmidt composition operators onH 2.Indiana University Mathematics Journal 23 (1973), 471–496.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Szpankowski, W., and Frieze, A. Greedy algorithms for the shortest common superstring that are asymptotically optimal. Preprint, 1998Google Scholar
  37. [37]
    Szpankowski, W., and Luczak, T. A suboptimal lossy date compression based on appoximate pattern matching.IEEE Transactions on Information Theory 43(5) (1997), 1439–1451.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    Tenenbaum, G.Introduction à la théorie analytique des nombres, vol. 13. Institut Élie Cartan, Nancy, 1990.Google Scholar
  39. [39]
    Vallée, B. Opérateurs de Ruelle-Mayer généralisés et analyse des algorithmes d’Euclide et de Gauss.Acta Arithmetica 81(2) (1997), 101–144.zbMATHMathSciNetGoogle Scholar
  40. [40]
    Vallée, B. Algorithms for computing signs of 2×2 determinants: dynamics and average-case algorithms,Proceedings of the 8th Annual European Symposium on Algorithms, ESA ’97, pp. 486–499. LNCS 1284. Springer-Verlag, Berlin, 1997.Google Scholar
  41. [41]
    Welsh, D.Codes and Cryptography. Oxford Science Publications, Clarendon Press, Oxford, 1989.Google Scholar

Copyright information

© Springer-Verlag New York Inc 2001

Authors and Affiliations

  • B. Vallée
    • 1
  1. 1.GREYC, Université de CaenCaenFrance

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