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.


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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