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Zbl 1091.37008
Ecalle, Jean; Vallet, Bruno
Intertwined mappings. (English. French summary)
[J] Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 3, 291-376 (2004). ISSN 0240-2963

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$(\bbfC,0)$ that are `entwined' in a group relation $W(f,g)=\text{id}$. 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 2000:: *37C05 Smooth mappings and diffeomorphisms
34M15 Algebraic aspects of differential equations in the complex domain
37E20 Universality, renormalization
22E99 Lie groups
37F50 Small divisors, rotation domains and linearization
37G05 Normal forms
58D99 Spaces and manifolds of mappings
32B10 Germs of analytic sets
32S65 Singularities of holomorphic vector fields

Keywords: local complex diffeomorphisms; bound groups of mappings; twins; mapping germs; identity-tangent mappings

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