Intertwined mappings. (English. French summary) 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\((\mathbb{C},0)\) 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:
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 34M15 | Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain |
| 37E20 | Universality and renormalization of dynamical systems |
| 22E99 | Lie groups |
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
| 37G05 | Normal forms for dynamical systems |
| 58D99 | Spaces and manifolds of mappings (including nonlinear versions of 46Exx) |
| 32B10 | Germs of analytic sets, local parametrization |
| 32S65 | Singularities of holomorphic vector fields and foliations |
Keywords:
local complex diffeomorphisms; bound groups of mappings; twins; mapping germs; identity-tangent mappingsReferences:
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