Extension of Chronological Calculus for Dynamical Systems on Manifolds

Abstract

We propose an extension of the Chronological Calculus, developed by Agrachev and Gamkrelidze for the case of $C^\infty$-smooth dynamical systems on finite-dimensional $C^\infty$-smooth manifolds, to the case of $C^m$-smooth dynamical systems and infinite-dimensional $C^m$-manifolds. Due to a relaxation in the underlying structure of the calculus, this extension provides a powerful computational tool without recourse to the theory of calculus in Fr\'echet spaces required by the classical Chronological Calculus. In addition, this extension accounts for flows of vector fields which are merely measurable in time. To demonstrate the utility of this extension, we prove a variant of Chow-Rashevskii theorem for infinite-dimensional manifolds.