## Generalized Morse sequences.(English)Zbl 0162.07201

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.

### MSC:

 37Axx Ergodic theory 28Dxx Measure-theoretic ergodic theory
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### References:

 [1] Gottschalk, W. H.: Almost periodic points with respect to transformation semigroups. Ann. of Math., II. Ser. 47, 762-766 (1946). · Zbl 0063.01713 [2] - and G. A. Hedlund: Topological dynamics. Amer. Math. Soc. Coll. Publ. 36, (1955). [3] Hahn, F. J., and Y. Katznelson: On the entropy of uniquely ergodic transformations. Trans. Amer. math. Soc. 126, 335-360 (1966). · Zbl 0191.21502 [4] Kakutani, S.: Ergodic theory of shift transformations. Proc. Fifth Berkeley Sympos. math. Statist. Probability II, 405-414 (1967). · Zbl 0217.38004 [5] Knoppp, K.: Infinite sequences and series. New York: Dover 1956. [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). · Zbl 0016.08604 [7] Morse, M., and G. A. Hedlund: Symbolic dynamics. Amer. J. Math. 60, 815-866 (1938). · Zbl 0019.33502 [8] ??: Unending chess, symbolic dynamics, and a problem in semigroups. Duke math. J. 11, 1-7 (1944). · Zbl 0063.04115 [9] Nemyckii, V. V., and V. V. Stepanov: Qualitative theory of differential equations. Princeton Math. Ser. 22 (1960). [10] Oxtoby, J. C.: Ergodic sets. Bull. Amer. math. Soc. 58, 116-136 (1952). · Zbl 0046.11504
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