Summary
Measure-theoretical and topological mixing properties of dynamical systems arising from substitutions are considered. It is shown that such systems are neither measure-theoretically strongly mixing, nor topologically strongly mixing of all orders. In the case of the substitution
it is shown that the corresponding dynamical system is measure-theoretically weakly but not strongly mixing, and topologically strongly mixing of order two but not of order three.
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Chacon, R.V.: Transformations having continuous spectrum. J. Math. Mech. 16, 399–415 (1966)
Coven, E.M., Keane, M.: The structure of substitution minimal sets. Trans. Amer. Math. Soc. 162, 89–102 (1971)
Dekking, F.M.: The spectrum of dynamical systems arising from substitutions of constant length. Preprint (1977)
Dekking, F.M., Keane, M., Michel, P.: Substitutions. Seminar manuscript, Rennes (1977)
Friedman, N.A.: Introduction to ergodic theory. New York: Van Nostrand-Reinholt 1970
Goodman, S., Marcus, B.: Topological mixing of higher degrees. Preprint (1977)
Gottschalk, W., Hedlund, G.A.: Topological dynamics. Amer. Math. Soc. Colloq. Publ. Vol. 36, Providence (1955)
Halmos, P.R.: Lectures on ergodic theory. New York: Chelsea 1956
Jacobs, K., Keane, M.: 0–1-Sequences of Toeplitz type. Z. Wahrscheinlichkeitstheorie verw. Gebiete 13, 123–131 (1969)
Kakutani, S.: Strictly ergodic symbolic dynamical systems. Proc. VI Berk. Sympos. Math. Statist. Probab. vol. II, 319–326. Univ. Calif. (1972)
Kakutani, S.: Examples of ergodic measure preserving transformations which are weakly mixing but not strongly mixing. SLN 318, 143–149 (1972)
Michel, P.: Stricte ergodicité d'ensembles minimaux de substitutions. C.R. Acad. Sci. Paris Sér. A-B 278, 811–813 (1974)
Neveu, J.: Sur les suites de Toeplitz. Z. Wahrscheinlichkeitstheorie verw. Gebiete 13, 132–134 (1969)
Petersen, K.E.: A topologically strongly mixing symbolic minimal set. Trans. Amer. Math. Soc. 148, 603–612 (1970)
Petersen, K.E., Shapiro, L.: Induced Flows. Trans. Amer. Math. Soc. 177, 375–389 (1973)
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Dekking, F.M., Keane, M. Mixing properties of substitutions. Z. Wahrscheinlichkeitstheorie verw Gebiete 42, 23–33 (1978). https://doi.org/10.1007/BF00534205
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DOI: https://doi.org/10.1007/BF00534205