A topologically strongly mixing symbolic minimal set
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- by K. E. Petersen PDF
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Abstract:
Recent papers by the author, Keynes and Robertson, and others have shown that weakly mixing minimal flows are objects of considerable interest, but examples of such flows, other than the horocycle flows, have been scarce. We give here a “machinal” construction of a bilateral sequence with entries from 0, 1 whose orbit closure is topologically strongly mixing and minimal. We prove in addition that the flow we obtain has entropy zero, is uniquely ergodic, and fails to be measure-theoretically strongly mixing.References
- R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
- Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
- 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
- Frank Hahn and Yitzhak Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc. 126 (1967), 335–360. MR 207959, DOI 10.1090/S0002-9947-1967-0207959-1 G. A. Hedlund, “Transformations commuting with the shift,” in Topological dynamics, J. Auslander and W. H. Gottschalk (Editors), Benjamin, New York, 1968, pp. 259-289. MR 38 #2762.
- Konrad Jacobs, Systèmes dynamiques Riemanniens, Czechoslovak Math. J. 20(95) (1970), 628–631 (French). MR 268876
- 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
- Harvey B. Keynes and James B. Robertson, Eigenvalue theorems in topological transformation groups, Trans. Amer. Math. Soc. 139 (1969), 359–369. MR 237748, DOI 10.1090/S0002-9947-1969-0237748-5
- 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
- John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
- K. E. Petersen, A topologically strongly mixing symbolic minimal set, Trans. Amer. Math. Soc. 148 (1970), 603–612. MR 259884, DOI 10.1090/S0002-9947-1970-0259884-8
- K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc. 24 (1970), 278–280. MR 250283, DOI 10.1090/S0002-9939-1970-0250283-7
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 148 (1970), 603-612
- MSC: Primary 54.82
- DOI: https://doi.org/10.1090/S0002-9947-1970-0259884-8
- MathSciNet review: 0259884