Explicit linearization of one-dimensional germs through tree-expansions

Abstract : We explain Ecalle's ''arbomould formalism'' in its simplest instance, showing how it allows one to give explicit formulas for the operators naturally attached to a germ of holomorphic map in one dimension. When applied to the classical linearization problem of non-resonant germs, which contains the well-known difficulties due to the so-called small divisor phenomenon, this elegant and concise tree formalism yields compact formulas, from which one easily recovers the classical analytical results of convergence of the solution under suitable arithmetical conditions on the multiplier. We rediscover this way Yoccoz's lower bound for the radius of convergence of the linearization and can even reach a global regularity result with respect to the multiplier (C^1-holomorphy) which improves on Carminati-Marmi's result. An appendix is devoted to the relationship between Ecalle's formalism and other algebraic constructions involving trees (the Connes-Kreimer Hopf algebra, a pre-Lie algebra, and representations thereof).
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Frédéric Fauvet, Frédéric Menous, David Sauzin. Explicit linearization of one-dimensional germs through tree-expansions. 2014. ⟨hal-01053805v2⟩

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