Ecalle's dimorphic transportation and Brown's lifting
Hanamichi Kawamura
TL;DR
The paper embeds Brown's lifting of linearized double shuffle solutions into Ecalle's dimorphic transportation within the flexion framework, proving that for any $f$ in the linearized solution space $\mathfrak{ls}_{\cQ}$ the lifted element $\chi_{B}(f)$ is a double shuffle solution in $\mathfrak{ds}_{\cQ}$. Central to the construction is identifying Brown's lifting with the adjoint action $\mathsf{adari}(\mathsf{par})$ up to the swap/anti conjugation, and building this bridge via the dilator calculus associated with $\psi_{0}$, a polar dilator lying in $\mathfrak{ds}_{\cQ}$. The work develops the necessary flexion and riemannian-like (Ihara) structures, shows how dilators interact with derivations and adjoint actions, and culminates in a precise, recursive formula that demonstrates $\chi_{B}(f)^{\sharp}=\mathsf{adari}(\mathsf{par})(f^{\sharp})$, thereby establishing Brown’s lifting as a dimorphic transportation. The results not only confirm conjectures about Brown’s lifting but also open a pathway to generalizing to other liftings $\chi_{\psi}$ via the same dimorphic framework.
Abstract
Brown's lifting procedure shows that one can solve the double shuffle equations modulo products from solutions of linearized ones. In this paper, we reveal that it is described in the framework of Ecalle's dimorphic transportation.