O'Farrell, Anthony G. and Roginskaya, Maria
Reducing Conjugacy in the full diffeomorphism
group of R to conjugacy in the subgroup of
Journal of Mathematical Sciences, 158 (6).
Let Diffeo = Diffeo(R) denote the group of infinitely-differentiable diffeomorphisms
of the real line R, under the operation of composition, and let Diffeo+
be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving
diffeomorphisms. We show how to reduce the problem of determining whether
or not two given elements f, g ∈ Diffeo are conjugate in Diffeo to associated
conjugacy problems in the subgroup Diffeo+. The main result concerns the
case when f and g have degree −1, and specifies (in an explicit and verifiable
way) precisely what must be added to the assumption that their (compositional)
squares are conjugate in Diffeo+, in order to ensure that f is conjugated to g by an element of Diffeo+. The methods involve formal power series, and results
of Kopell on centralisers in the diffeomorphism group of a half-open interval.
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