Multiple Lie Derivatives and Forests

TL;DR

This paper develops a comprehensive time expansion of the pull-back operator using multiple Lie derivatives, providing explicit estimates for higher order derivatives through combinatorial structures like forests and Dyck polynomials.

Contribution

It introduces a complete expansion of the pull-back operator in terms of multiple Lie derivatives and offers explicit estimates for derivatives using combinatorial methods.

Findings

Complete time expansion of the pull-back operator in terms of multiple Lie derivatives.

Explicit estimates for higher order covariant derivatives using Dyck polynomials.

Novel combinatorial approach involving rooted labeled forests and Dyck polynomials.

Abstract

We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for higher order covariant derivatives of multiple Lie derivatives acting on smooth endomorphism sections of the tangent bundle of a manifold. We assume the covariant derivative to be torsion free. The estimate is given in terms of Dyck polynomials. The proof uses a new result on the combinatorics of rooted labeled ordered forests and Dyck polynomials.