Publicacions Matemàtiques

Reversibility in the Diffeomorphism Group of the Real Line

Anthony G. O’Farrell and Ian Short

Full-text: Open access

Abstract

An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.

Article information

Source
Publ. Mat., Volume 53, Number 2 (2009), 401-415.

Dates
First available in Project Euclid: 20 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.pm/1248095661

Mathematical Reviews number (MathSciNet)
MR2543857

Zentralblatt MATH identifier
1178.37022

Subjects
Primary: 37C05: Smooth mappings and diffeomorphisms 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification

Keywords
Diffeomorphism reversible involution conjugacy

Citation

O’Farrell, Anthony G.; Short, Ian. Reversibility in the Diffeomorphism Group of the Real Line. Publ. Mat. 53 (2009), no. 2, 401--415. https://projecteuclid.org/euclid.pm/1248095661


Export citation

References

  • A. B. Calica, Reversible homeomorphisms of the real line, Pacific J. Math. 39 (1971), 79\Ndash87.
  • R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc. 218 (1976), 89\Ndash113.
  • N. J. Fine and G. E. Schweigert, On the group of homeomorphisms of an arc, Ann. of Math. (2) 62 (1955), 237\Ndash253.
  • W. Jarczyk, Reversible interval homeomorphisms, J. Math. Anal. Appl. 272(2) (2002), 473\Ndash479.
  • N. Kopell, Commuting diffeomorphisms, in: “Global Analysis”, Proc. Sympos. Pure Math. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 165\Ndash184.
  • J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Time-reversal symmetry in dynamical systems (Coventry, 1996), Phys. D 112(1–2) (1998), 1\Ndash39.
  • J. Lubin, Non-Archimedean dynamical systems, Compositio Math. 94(3) (1994), 321\Ndash346.
  • A. G. O'Farrell, Conjugacy, involutions, and reversibility for real homeomorphisms, Irish Math. Soc. Bull. 54 (2004), 41\Ndash52.
  • A. G. O'Farrell, Composition of involutive power series, and reversible series, Comput. Methods Funct. Theory 8(1–2) (2008), 173\Ndash193.
  • A. G. O'Farrell and M. Roginskaya, Reducing conjugacy in the full diffeomorphism group of $\mathbb{R}$ to conjugacy in the subgroup of order preserving maps, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (Russian) 360 (2008), 231\Ndash237; translation in: J. Math. Sci. (N.Y.) 158 (2009), 895\Ndash898.
  • S. Sternberg, Local $C\sp{n}$ transformations of the real line, Duke Math. J. 24 (1957), 97\Ndash102.
  • S. W. Young, The representation of homeomorphisms on the interval as finite compositions of involutions, Proc. Amer. Math. Soc. 121(2) (1994), 605\Ndash610.