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Generalized Morse sequences

  • M. Keane
Article

Summary

A method for construction of almost periodic points in the shift space on two symbols is developed, and a necessary and sufficient condition is given for the orbit closure of such a point to be strictly ergodic. Points satisfying this condition are called generalized Morse sequences. The spectral properties of the shift operator in strictly ergodic systems arising from generalized Morse sequences are investigated. It is shown that under certain broad regularity conditions both the continuous and discrete parts of the spectrum are non-trivial. The eigenfunctions and eigenvalues are calculated. Using the results, given any subgroup of the group of roots of unity, a generalized Morse sequence can be constructed whose continuous spectrum is non-trivial and whose eigenvalue group is precisely the given group. New examples are given for almost periodic points whose orbit closure is not strictly ergodic.

Keywords

Stochastic Process Probability Theory Spectral Property Mathematical Biology Continuous Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

  1. 1.
    Gottschalk, W. H.: Almost periodic points with respect to transformation semigroups. Ann. of Math., II. Ser. 47, 762–766 (1946).Google Scholar
  2. 2.
    - and G. A. Hedlund: Topological dynamics. Amer. Math. Soc. Coll. Publ. 36, (1955).Google Scholar
  3. 3.
    Hahn, F. J., and Y. Katznelson: On the entropy of uniquely ergodic transformations. Trans. Amer. math. Soc. 126, 335–360 (1966).Google Scholar
  4. 4.
    Kakutani, S.: Ergodic theory of shift transformations. Proc. Fifth Berkeley Sympos. math. Statist. Probability II, 405–414 (1967).Google Scholar
  5. 5.
    Knoppp, K.: Infinite sequences and series. New York: Dover 1956.Google Scholar
  6. 6.
    Krylof, N., et N. Bogoliouboff: La theorie generale de la mesure dans son application a l'etude des systemes dynamiques de la mechanique non lineaire. A. of Math., II. Ser. 38, 65–113 (1937).Google Scholar
  7. 7.
    Morse, M., and G. A. Hedlund: Symbolic dynamics. Amer. J. Math. 60, 815–866 (1938).Google Scholar
  8. 8.
    ——: Unending chess, symbolic dynamics, and a problem in semigroups. Duke math. J. 11, 1–7 (1944).Google Scholar
  9. 9.
    Nemyckii, V. V., and V. V. Stepanov: Qualitative theory of differential equations. Princeton Math. Ser. 22 (1960).Google Scholar
  10. 10.
    Oxtoby, J. C.: Ergodic sets. Bull. Amer. math. Soc. 58, 116–136 (1952).Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • M. Keane
    • 1
  1. 1.Dept. of MathematicsYale UniversityNew Haven

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