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Homomorphisms of symbolic dynamical systems

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Abstract

The densities of finite blocks of symbols can be computed for points in any substitution minimal set arising from a substitution of constant length and also for points in any Sturmian minimal set. It follows from the computations that all Sturmian minimal sets and all substitution minimal sets are strictly ergodic and have topological entropy zero. It also follows that no Sturmian minimal set is the homomorphic image of a substitution minimal set. It can also be shown that no non-trivial substitution minimal set is powerfully (totally) minimal, and it follows that no non-trivial substitution minimal set is the homomorphic image of a Sturmian minimal set.

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References

  1. R. L. Adler, A. G. Konheim andM. H. McAndrew, Topological Entropy,Trans. Amer. Math. Soc. 114 (1965), 309–319.

    Google Scholar 

  2. P. Billingsley,Ergodic Theory and Information, John Wiley and Sons, New York, 1965.

    Google Scholar 

  3. E. M. Coven andM. S. Keane, The structure of substitution minimal sets, to appear.

  4. J. L. Doob,Stochastic Processes, John Wiley and Sons, New York, 1953.

    Google Scholar 

  5. R. Ellis andW. H. Gottschalk, Homomorphisms of transformation groups,Trans. Amer. Math. Soc. 94 (1960), 258–271.

    Google Scholar 

  6. W. H. Gottschalk, Substitution minimal sets,Trans. Amer. Math. Soc. 109 (1963), 467–491.

    Google Scholar 

  7. W. H. Gottschalk andG. A. Hedlund,Topological Dynamics, Amer. Math. Soc. Colloq. Publ., Vol. 36, American Mathematical Society, Providence, 1955.

    Google Scholar 

  8. F. J. Hahn andY. Katznelson, On the entropy of uniquely ergodic transformations,Trans. Amer. Math. Soc. 126 (1967), 335–360.

    Google Scholar 

  9. G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system,Math. Systems Theory 3 (1969), 320–375.

    Google Scholar 

  10. G. A. Hedlund, Sturmian minimal sets,Amer. J. Math. 66 (1944), 605–620.

    Google Scholar 

  11. M. Morse andG. A. Hedlund, Symbolic dynamics,Amer. J. Math. 60 (1938), 815–866.

    Google Scholar 

  12. M. Morse andG. A. Hedlund, Symbolic dynamics II. Sturmian trajectories,Amer. J. Math. 62 (1940), 1–42.

    Google Scholar 

  13. J. C. Oxtoby, Ergodic sets,Bull. Amer. Math. Soc. 58 (1952), 116–136.

    Google Scholar 

  14. H. R. Pitt, Some generalizations of the ergodic theorem,Proc. Cambridge Phil. Soc. 38 (1942), 325–343.

    Google Scholar 

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Authors and Affiliations

  1. New York University and Davidson College, 28036, Davidson, North Carolina, USA

    Benjamin G. Klein

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  1. Benjamin G. Klein
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Additional information

Most of the results in this paper were first obtained in the author's doctoral dissertation which was presented to the Faculty of the Graduate School of Yale University in 1968.

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Klein, B.G. Homomorphisms of symbolic dynamical systems. Math. Systems Theory 6, 107–122 (1972). https://doi.org/10.1007/BF01706082

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  • DOI: https://doi.org/10.1007/BF01706082

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