The tridendriform structure of a Magnus expansion

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a lihear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik-Plebanski-Strichartz formula for this logarithm.
Comment: To appear in Discrete and Continuous Dynamical Systems series A (AIMS). Partly supported by Agence Nationale de la Recherche, projet CARMA ANR-12-BS01-0017, and Ministerio de Econom{\i}a y Competitividad.MTM2011-23050