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**Reversibility in the diffeomorphism group of the real line.**
*(English)*
Zbl 1178.37022

The article deals with reversible and strongly reversible elements of two groups: the group \({\text{Diffeo}} \, ({\mathbb R})\) of infinitely differentiable homeomorphisms (diffeomorphisms) of \({\mathbb R}\) and its subgroup \({\text{Diffeo}}^+ \, ({\mathbb R})\) of order preserving diffeomorphisms. An element \(f\) at such a group is called reversible (strongly reversible) if \(f^{-1} = h^{-1}fh\) for any \(h\) from the group (for any involution from the group). The main results are the following: (1) An element of \(\text{Diffeo} \, ({\mathbb R})\) is reversible if and only if it is conjugate to a map \(f\) from \(\text{Diffeo} \, ({\mathbb R})\) that fixed each integer and satisfies the equation \(f(x + 1) = f^{-1}(x) + 1\), \(x \in {\mathbb R}\); (2) Each reversible element of \(\text{Diffeo} \, ({\mathbb R})\) is either (i) strongly reversible, or (ii) an reversible (in \(\text{Diffeo}^+ \, ({\mathbb R})\)) element of \({\text{Diffeo}}^+ \, ({\mathbb R})\). It is also considered the problem about representation of elements of the groups \(\text{Diffeo} \, ({\mathbb R})\) as the composite of reversible diffeomorphisms or involutions. More precisely, it is proved that each element of \(\text{Diffeo} \, ({\mathbb R})\) can be expressed as a compsite of four involutions and each element of \(\text{Diffeo}^+ \,({\mathbb R})\) can be expressed as a composite of four reversible diffeomorphisms.

Reviewer: Peter Zabreiko (Minsk)

### MSC:

37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |

37E05 | Dynamical systems involving maps of the interval |

37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |

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\textit{A. G. O'Farrell} and \textit{I. Short}, Publ. Mat., Barc. 53, No. 2, 401--415 (2009; Zbl 1178.37022)

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