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Keane, M.

Generalized Morse sequences. (English) Zbl 0162.07201

Z. Wahrscheinlichkeitstheor. Verw. Geb. 10, 335-353 (1968).
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.
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Cited in 4 Reviews
Cited in 68 Documents

MSC:

37Axx Ergodic theory
28Dxx Measure-theoretic ergodic theory

Keywords:

construction of almost periodic points; shift space on two symbols; strictly ergodic orbit closure; generalized Morse sequences; spectral properties of shift operator; eigenfunctions and eigenvalues
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\textit{M. Keane}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 10, 335--353 (1968; Zbl 0162.07201)
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References:

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