Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows

T. Downarowicz; Y. Lacroix

Studia Mathematica (1998)

  • Volume: 130, Issue: 2, page 149-170
  • ISSN: 0039-3223

Abstract

top
Let ( Z , T Z ) be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of ( Z , T Z ) is Borel isomorphic to an almost 1-1 extension of ( Z , T Z ) . Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz flow.

How to cite

top

Downarowicz, T., and Lacroix, Y.. "Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows." Studia Mathematica 130.2 (1998): 149-170. <http://eudml.org/doc/216549>.

@article{Downarowicz1998,
abstract = {Let $(Z,T_Z)$ be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of $(Z,T_Z)$ is Borel isomorphic to an almost 1-1 extension of $(Z,T_Z)$. Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz flow.},
author = {Downarowicz, T., Lacroix, Y.},
journal = {Studia Mathematica},
keywords = {almost 1-1 extension; invariant measure; isomorphism; Toeplitz flow; measure isomorphism; topological extension; subshift; almost extension; ergodic Toeplitz flow},
language = {eng},
number = {2},
pages = {149-170},
title = {Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows},
url = {http://eudml.org/doc/216549},
volume = {130},
year = {1998},
}

TY - JOUR
AU - Downarowicz, T.
AU - Lacroix, Y.
TI - Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows
JO - Studia Mathematica
PY - 1998
VL - 130
IS - 2
SP - 149
EP - 170
AB - Let $(Z,T_Z)$ be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of $(Z,T_Z)$ is Borel isomorphic to an almost 1-1 extension of $(Z,T_Z)$. Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz flow.
LA - eng
KW - almost 1-1 extension; invariant measure; isomorphism; Toeplitz flow; measure isomorphism; topological extension; subshift; almost extension; ergodic Toeplitz flow
UR - http://eudml.org/doc/216549
ER -

References

top
  1. [A] J. Auslander, Minimal Flows and Their Extensions, North-Holland Math. Stud. 153, North-Holland, 1988. Zbl0654.54027
  2. [B-G-K] F. Blanchard, E. Glasner and J. Kwiatkowski, Minimal self-joinings and positive topological entropy, Monatsh. Math. 120 (1995), 205-222. Zbl0859.54027
  3. [B-K1] W. Bułatek and J. Kwiatkowski, The topological centralizers of Toeplitz flows and their Z 2 -extensions, Publ. Math. 34 (1990), 45-65. Zbl0731.54027
  4. [B-K2] W. Bułatek and J. Kwiatkowski, Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers, Studia Math. 103 (1992), 133-142. Zbl0816.58028
  5. [D-G-S] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, Berlin, 1976. Zbl0328.28008
  6. [D-K] M. Denker and M. Keane, Almost topological dynamical systems, Israel J. Math. 34 (1979), 139-160. Zbl0441.28008
  7. [D1] T. Downarowicz, A minimal 0-1 subshift with noncompact set of invariant measures, Probab. Theory Related Fields 79 (1988), 29-35 Zbl0637.28014
  8. [D2] T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 (1991), 241-256. Zbl0746.58047
  9. [D3] T. Downarowicz, The Royal Couple conceals their mutual relationship - A noncoalescent Toeplitz flow, ibid. 97 (1997), 239-252. Zbl0893.54032
  10. [D-1] T. Downarowicz and A. Iwanik, Quasi-uniform convergence in compact dynamical systems, Studia Math. 89 (1998), 11-25. 
  11. [D-K-L] T. Downarowicz, J. Kwiatkowski and Y. Lacroix, A criterion for Toeplitz flows to be topologically isomorphic and applications, Colloq. Math. 68 (1995), 219-228. Zbl0820.28009
  12. [D-L] T. Downarowicz and Y. Lacroix, A non-regular Toeplitz flow with preset pure point spectrum, Studia Math. 120 (1996), 235-246. Zbl0888.28009
  13. [E] E. Eberlein, Toeplitzfolgen und Gruppentranslationen, Thesis, Erlangen, 1970. 
  14. [F-K-M] S. Férenczi, J. Kwiatkowski and C. Mauduit, A density theorem for (multiplicity, rank) pairs, J. Anal. Math. 65 (1995), 45-75. Zbl0833.28010
  15. [F-W] H. Furstenberg and B. Weiss, On almost 1-1 extensions, Israel J. Math. 65 (1989), 311-322. Zbl0676.28010
  16. [Ga-H] M. Garsia and G. A. Hedlund, The structure of minimal sets, Bull. Amer. Math. Soc. 54 (1948), 954-964. Zbl0032.36002
  17. [G-H] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 36, 1995. Zbl0067.15204
  18. [H-R] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol. I, Springer, Berlin, 1963. 
  19. [I] A. Iwanik, Toeplitz flows with pure point spectrum, Studia Math. 118 (1996), 27-35. Zbl0888.28008
  20. [I-L] A. Iwanik and Y. Lacroix, Some constructions of strictly ergodic non-regular Toeplitz flows, ibid. 110 (1994), 191-203 Zbl0810.28009
  21. [J-K] K. Jacobs and M. Keane, 0-1 sequences of Toeplitz type, Z. Wahrsch. Verw. Gebiete 13 (1969), 123-131. Zbl0195.52703
  22. [K-L] J. Kwiatkowski and Y. Lacroix, Rank and weak closure theorem, II, preprint. Zbl1198.37009
  23. [L] M. Lemańczyk, Toeplitz Z 2 -extensions, Ann. Inst. H. Poincaré 24 (1998), 1-43. 
  24. [M-P] N. Markley and M. E. Paul, Almost automorphic symbolic minimal sets without unique ergodicity, Israel J. Math. 34 (1979), 259-272. Zbl0463.54039
  25. [O] J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136. Zbl0046.11504
  26. [W] B. Weiss, Strictly ergodic models for dynamical systems, ibid. 13 (1985), 143-146. Zbl0615.28012
  27. [Wi] S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete 67 (1984), 95-107. Zbl0584.28007

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.