Some constructions of strictly ergodic non-regular Toeplitz flows

A. Iwanik; Y. Lacroix

Studia Mathematica (1994)

  • Volume: 110, Issue: 2, page 191-203
  • ISSN: 0039-3223

Abstract

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We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.

How to cite

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Iwanik, A., and Lacroix, Y.. "Some constructions of strictly ergodic non-regular Toeplitz flows." Studia Mathematica 110.2 (1994): 191-203. <http://eudml.org/doc/216108>.

@article{Iwanik1994,
abstract = {We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.},
author = {Iwanik, A., Lacroix, Y.},
journal = {Studia Mathematica},
keywords = {strict ergodicity; regularity; cyclic approximation; Toeplitz flows; shift dynamical system},
language = {eng},
number = {2},
pages = {191-203},
title = {Some constructions of strictly ergodic non-regular Toeplitz flows},
url = {http://eudml.org/doc/216108},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Iwanik, A.
AU - Lacroix, Y.
TI - Some constructions of strictly ergodic non-regular Toeplitz flows
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 2
SP - 191
EP - 203
AB - We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.
LA - eng
KW - strict ergodicity; regularity; cyclic approximation; Toeplitz flows; shift dynamical system
UR - http://eudml.org/doc/216108
ER -

References

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  1. [Co-Fo-Si] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1982. 
  2. [Do-Iw] T. Downarowicz and A. Iwanik, Quasi-uniform convergence in compact dynamical systems, Studia Math. 89 (1988), 11-25. 
  3. [Do-Kw-La] T. Downarowicz, J. Kwiatkowski and Y. Lacroix, A criterion for Toeplitz flows to be isomorphic and applications, preprint. Zbl0820.28009
  4. [Fu] H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. Math. 83 (1961), 573-601. Zbl0178.38404
  5. [He-Ro] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, I, Springer, 1963. 
  6. [Iw] A. Iwanik, Approximation by periodic transformations and diophantine approximation of the spectrum, preprint. 
  7. [Ja-Ke] K. Jacobs and M. Keane, 0-1 sequences of Toeplitz type, Z. Wahrsch. Verw. Gebiete 13 (1969), 123-131. Zbl0195.52703
  8. [Ke] M. Keane, Generalized Morse sequences, ibid. 10 (1968), 335-353. Zbl0162.07201
  9. [La1] Y. Lacroix, Contribution à l'étude des suites de Toeplitz et numération en produit infini, Thesis, Université de Provence, 1992. 
  10. [La2] Y. Lacroix, Metric properties of generalized Cantor products, Acta Arith. 63 (1993), 61-77. Zbl0774.11042
  11. [Le] M. Lemańczyk, Ergodic Z 2 -extensions over rational pure point spectrum, category and homomorphisms, Compositio Math. 63 (1987), 63-81. Zbl0629.28013
  12. [Ne] D. Newton, On the entropy of certain classes of skew-product transformations, Proc. Amer. Math. Soc. 21 (1969), 722-726. Zbl0174.09201
  13. [Ox] J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136. Zbl0046.11504
  14. [Wi] S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete 67 (1984), 95-107. Zbl0584.28007

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