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Combinatorics of Poincaré’s and Schröder’s equations

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Resurgence, Physics and Numbers

Part of the book series: CRM Series ((CRMSNS,volume 20))

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Abstract

We investigate the combinatorial properties of the functional equation ϕ[h(z)] = h(qz) for the conjugation of a formal diffeomorphism ϕ of ℂ to its linear part z ↦ qz. This is done by interpreting the functional equation in terms of symmetric functions, and then lifting it to noncommutative symmetric functions. We describe explicitly the expansion of the solution in terms of plane trees and prove that its expression on the ribbon basis has coefficients in ℕ[q] after clearing the denominators (q) n . We show that the conjugacy equation can be lifted to a quadratic fixed point equation in the free triduplicial algebra on one generator. This can be regarded as a q-deformation of the duplicial interpretation of the noncommutative Lagrange inversion formula. Finally these calculations are interpreted in terms of the group of the operad of Stasheff polytopes, and are related to Ecalle’s arborified expansion by means of morphisms between various Hopf algebras of trees.

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Authors
  1. Frédéric Menous
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  2. Jean-Christophe Novelli
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  3. Jean-Yves Thibon
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Editor information

Frédéric Fauvet Dominique Manchon Stefano Marmi David Sauzin

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© 2017 Scuola Normale Superiore Pisa

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Menous, F., Novelli, JC., Thibon, JY. (2017). Combinatorics of Poincaré’s and Schröder’s equations. In: Fauvet, F., Manchon, D., Marmi, S., Sauzin, D. (eds) Resurgence, Physics and Numbers. CRM Series, vol 20. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-613-1_8

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