Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

TOPOLOGICAL CONJUGACY OF DISJOINT FLOWS ON THE CIRCLE

  • Published : 2002.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let $F={F^v:S^1->S^1,v\in\; V$ and $g={G^v:S^1->S^1,v\in\; V$ be disjoint flows defined on the unit circle $S^1$, that is such flows that each their element either is the identity mapping or has no fixed point ((V, +) is a 2-divisible nontrivial abelian group). The aim of this paper is to give a necessary and sufficient codition for topological conjugacy of disjoint flows i.e., the existence of a homeomorphism $\Gamma:S^1->S^1$ satisfying $$\Gamma\circ\ F^v=G^v\circ\Gamma,\; v\in\; V$$ Moreover, under some further restrictions, we determine all such homeomorphisms.

Keywords

References

  1. L. Alseda, J. Llibre, and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993.
  2. J. S. Bae, K. J. Min, D. H. Sung, and S. K. Yang, Positively equicontinuous flows are topologically conjugate to rotation flows, Bull. Korean Math. Soc. 36 (1999), no. 4, 707-716.
  3. M. Bajger, On the structure of some flows on the unit circle, Aequationes Math. 55 (1998), no. 1-2, 106-121. https://doi.org/10.1007/s000100050023
  4. M. Bajger and M. C. Zdun, On rational flows of continuous functions, Iteration theory (Batschuns, 1992), 265-276, World Sci. Publishing, River Edge, NJ, 1996.
  5. L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992.
  6. K. Cieplinski, On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups, Publ. Math. Debrecen 55 (1999), no. 3-4, 363-383.
  7. K. Cieplinski, On conjugacy of disjoint iteration groups on the unit circle, European Conference on Iteration Theory (Muszyna-Zlockie, 1998), Ann. Math. Sil. 13 (1999), 103-118.
  8. K. Cieplinski, The rotation number of the composition of homeomorphisms, Rocznik Nauk.-Dydakt. AP w Krakowie. Prace Mat. 17 (2000), 83-87.
  9. K. Cieplinski, The structure of disjoint iteration groups on the circle, Czechoslovak, Math. J . (to appear) https://doi.org/10.1023/B:CMAJ.0000027254.04824.0c
  10. K. Cieplinski and M. C. Zdun, On semi-conjugacy equation for homeomorphisms of the circle, Functional Equations-Results and Advances, Kluwer Academic Publishers, Boston, Dordrecht, London (to appear)
  11. I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic theory, Grundlehren der Mathematischen Wissenschaften, 245, Springer-Verlag, New York, 1982.
  12. Z. Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971.
  13. C. T. C. Wall, A geometric introduction to topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972.
  14. M. C. Zdun, On embedding of homeomorphisms of the circle in a continuous flow, Iteration theory and its functional equations (Lochau, 1984), 218-231, Lecture Notes in Math., 1163, Springer, Berlin, 1985. https://doi.org/10.1007/BFb0076436
  15. M. C. Zdun, On some invariants of conjugacy of disjoint iteration groups, Results Math. 26 (1994), no. 3-4, 403-410. https://doi.org/10.1007/BF03323067

Cited by

  1. Recent results on iteration theory: iteration groups and semigroups in the real case vol.87, pp.3, 2014, https://doi.org/10.1007/s00010-013-0186-x
  2. General Construction of Non-Dense Disjoint Iteration Groups on the Circle vol.55, pp.4, 2005, https://doi.org/10.1007/s10587-005-0088-8
  3. Schröder equation and commuting functions on the circle vol.342, pp.1, 2008, https://doi.org/10.1016/j.jmaa.2007.12.031
  4. The Structure of Disjoint Iteration Groups on the Circle vol.54, pp.1, 2004, https://doi.org/10.1023/B:CMAJ.0000027254.04824.0c