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

O’Farrell, Anthony G.; Short, Ian

Publ. Mat., Barc. 53, No. 2, 401-415 (2009).

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

Keywords:

diffeomorphism; reversible; involution; conjugacy

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