The structure of disjoint iteration groups on the circle

Krzysztof Ciepliński

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 1, page 131-153
  • ISSN: 0011-4642

Abstract

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The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle 𝕊 1 , that is, families ℱ = { F v 𝕊 1 ⟶ 𝕊 1 v ∈ V } of homeomorphisms such that F v 1 ∘ F v 2 = F v 1 + v 2 , v 1 , v 2 ∈ V , and each F v either is the identity mapping or has no fixed point ( ( V , + ) is an arbitrary 2 -divisible nontrivial (i.e., c a r d V > 1 ) abelian group).

How to cite

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Ciepliński, Krzysztof. "The structure of disjoint iteration groups on the circle." Czechoslovak Mathematical Journal 54.1 (2004): 131-153. <http://eudml.org/doc/30844>.

@article{Ciepliński2004,
abstract = {The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle $\{\mathbb \{S\}^1\}$, that is, families $\{\mathcal \{F\}\}=\lbrace F^\{v\}\:\{\mathbb \{S\}^1\}\longrightarrow \{\mathbb \{S\}^1\}\; v\in V\rbrace $ of homeomorphisms such that \[ F^\{v\_\{1\}\}\circ F^\{v\_\{2\}\}=F^\{v\_\{1\}+v\_\{2\}\},\quad v\_1, v\_2\in V, \] and each $F^\{v\}$ either is the identity mapping or has no fixed point ($(V, +)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathop \{\mathrm \{c\}ard\}V>1$) abelian group).},
author = {Ciepliński, Krzysztof},
journal = {Czechoslovak Mathematical Journal},
keywords = {(disjoint; non-singular; singular; non-dense; dense; discrete) iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations; iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations},
language = {eng},
number = {1},
pages = {131-153},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The structure of disjoint iteration groups on the circle},
url = {http://eudml.org/doc/30844},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Ciepliński, Krzysztof
TI - The structure of disjoint iteration groups on the circle
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 131
EP - 153
AB - The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${\mathbb {S}^1}$, that is, families ${\mathcal {F}}=\lbrace F^{v}\:{\mathbb {S}^1}\longrightarrow {\mathbb {S}^1}\; v\in V\rbrace $ of homeomorphisms such that \[ F^{v_{1}}\circ F^{v_{2}}=F^{v_{1}+v_{2}},\quad v_1, v_2\in V, \] and each $F^{v}$ either is the identity mapping or has no fixed point ($(V, +)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathop {\mathrm {c}ard}V>1$) abelian group).
LA - eng
KW - (disjoint; non-singular; singular; non-dense; dense; discrete) iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations; iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations
UR - http://eudml.org/doc/30844
ER -

References

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  1. Combinatorial Dynamics and Entropy in Dimension One. Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co. Inc., River Edge, 1993. (1993) MR1255515
  2. 10.1007/s000100050023, Aequationes Math. 55 (1998), 106–121. (1998) Zbl0891.39017MR1600588DOI10.1007/s000100050023
  3. On rational flows of continuous functions, Iteration Theory (Batschuns, 1992), World Sci. Publishing, River Edge, 1996, pp. 265–276. (1996) MR1442291
  4. Dynamics in One Dimension. Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992. (1992) MR1176513
  5. Éléments de mathématique. Topologie générale, Hermann, Paris, 1971. (1971) Zbl0249.54001MR0358652
  6. On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups, Publ. Math. Debrecen 55 (1999), 363–383. (1999) MR1721896
  7. On conjugacy of disjoint iteration groups on the unit circle, Ann. Math. Sil. 13 (1999), 103–118. (1999) MR1735195
  8. The rotation number of the composition of homeomorphisms, Rocznik Nauk.—Dydakt. AP w Krakowie. Prace Mat. 17 (2000), 83–87. (2000) MR1817486
  9. 10.4134/BKMS.2002.39.2.333, Bull. Korean Math. Soc. 39 (2002), 333–346. (2002) MR1904668DOI10.4134/BKMS.2002.39.2.333
  10. 10.1142/S0218127403007709, Internat. J.  Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 1883–1888. (2003) MR2015635DOI10.1142/S0218127403007709
  11. Ergodic Theory. Grundlehren der Mathematischen Wissenschaften, 245, Springer-Verlag, New York, 1982. (1982) MR0832433
  12. Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The M.I.T. Press Cambridge, Massachusetts-London, 1971. (1971) Zbl0246.58012MR0649788
  13. A Geometric Introduction to Topology, Addison-Wesley Publishing Co., Reading, Massachusetts-London-Don Mills, 1972. (1972) MR0478128
  14. An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. (1982) MR0648108
  15. On embedding of homeomorphisms of the circle in a continuous flow, Iteration theory and its functional equations (Lochau, 1984). Lecture Notes in Math., 1163, Springer, Berlin, 1985, pp. 218–231. (1985) Zbl0616.54037MR0829776
  16. On the embeddability of commuting functions in a flow, Selected topics in functional equations and iteration theory (Graz, 1991). Grazer Math. Ber., 316, Karl-Franzens-Univ. Graz, Graz, 1992, pp. 201–212. (1992) Zbl0797.39007MR1226473
  17. On the orbits of disjoint groups of continuous functions, Rad. Mat. 8 (1992/96), 95–104. (1992/96) MR1477887
  18. 10.1007/BF01833995, Aequationes Math. 46 (1993), 19–37. (1993) Zbl0801.39005MR1220719DOI10.1007/BF01833995
  19. 10.1007/BF03323067, Results Math. 26 (1994), 403–410. (1994) Zbl0830.39009MR1300626DOI10.1007/BF03323067

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