The structure of generalized Morse minimal sets on $n$ symbols
HTML articles powered by AMS MathViewer
- by John C. Martin PDF
- Trans. Amer. Math. Soc. 232 (1977), 343-355 Request permission
Abstract:
A class of bisequences on n symbols is constructed which includes the generalized Morse sequences introduced by Keane. Those which give rise to strictly ergodic sets are characterized, and the spectrum of the shift operator on these systems is investigated. It is shown that in certain cases the shift operator has partly discrete and partly continuous spectrum. The theorems generalize results of Keane on generalized Morse sequences and a theorem of Kakutani regarding a particular strictly transitive sequence on four symbols. Another special case yields information on the spectrum of certain substitution minimal sets.References
- Ethan M. Coven and Michael S. Keane, The structure of substitution minimal sets, Trans. Amer. Math. Soc. 162 (1971), 89–102. MR 284995, DOI 10.1090/S0002-9947-1971-0284995-1
- W. H. Gottschalk, Substitution minimal sets, Trans. Amer. Math. Soc. 109 (1963), 467–491. MR 190915, DOI 10.1090/S0002-9947-1963-0190915-6
- Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810, DOI 10.1090/coll/036
- Shizuo Kakutani, Ergodic theory of shift transformations, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 405–414. MR 0227358
- M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 335–353. MR 239047, DOI 10.1007/BF00531855
- Benjamin G. Klein, Homomorphisms of symbolic dynamical systems, Math. Systems Theory 6 (1972), 107–122. MR 305383, DOI 10.1007/BF01706082 P. Michel, Sur les ensembles minimaux engendres par les substitutions de longueur non constante, Thesis, University of Rennes, 1975.
- Harold Marston Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), no. 1, 84–100. MR 1501161, DOI 10.1090/S0002-9947-1921-1501161-8
- Marston Morse and Gustav A. Hedlund, Symbolic Dynamics, Amer. J. Math. 60 (1938), no. 4, 815–866. MR 1507944, DOI 10.2307/2371264
- John C. Martin, Substitution minimal flows, Amer. J. Math. 93 (1971), 503–526. MR 300261, DOI 10.2307/2373391
- John C. Martin, Generalized Morse sequences on $n$ symbols, Proc. Amer. Math. Soc. 54 (1976), 379–383. MR 391058, DOI 10.1090/S0002-9939-1976-0391058-3
- John C. Martin, Limits of successive convolutions, Proc. Amer. Math. Soc. 59 (1976), no. 1, 52–54. MR 412734, DOI 10.1090/S0002-9939-1976-0412734-X
- John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 232 (1977), 343-355
- MSC: Primary 28A65
- DOI: https://doi.org/10.1090/S0002-9947-1977-0463400-1
- MathSciNet review: 0463400