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Intertwined mappings. (English) Zbl 1091.37008
Summary: We show that, contrary to expectations, there exist pairs of formal and even analytic, non-commuting and non-elementary (neither algebraic nor algebraic-differential) mapping germs in Diff(,0) that are ‘entwined’ in a group relation W(f,g)=id. In the case of identity-tangent mappings, ‘twins’ exhibit, rather than analyticity, generic divergence, but of a particularly interesting sort: resurgent, accelero-summable, and with simple alien derivatives.
MSC:
37C05Smooth mappings and diffeomorphisms
34M15Algebraic aspects of ODE in the complex domain
37E20Universality, renormalization
22E99Lie groups
37F50Small divisors, rotation domains and linearization; Fatou and Julia sets
37G05Normal forms
58D99Spaces and manifolds of mappings
32B10Germs of analytic sets, local parametrization
32S65Singularities of holomorphic vector fields and foliations