Publicacions Matemàtiques

Reversibility in the Diffeomorphism Group of the Real Line

Anthony G. O’Farrell and Ian Short

Full-text: Open access


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

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

First available in Project Euclid: 20 July 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Diffeomorphism reversible involution conjugacy


O’Farrell, Anthony G.; Short, Ian. Reversibility in the Diffeomorphism Group of the Real Line. Publ. Mat. 53 (2009), no. 2, 401--415.

Export citation


  • 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.