Geometric aspects of Young Integral: decomposition of flows

Abstract

In this paper we study geometric aspects of dynamics generated by Young differential equations (YDE) driven by $\alpha$-Hölder trajectories with $\alpha \in (1/2,1)$. We present a number of properties and geometrical constructions on this low regularity context: Young Itô geometrical formula, horizontal lift in principal fibre bundles, parallel transport, covariant derivative, development and anti-development, among others. Our main application here is a geometrical decomposition of flows generated by YDEs according to diffeomorphisms generated by complementary distributions (integrable or not). The proof of existence of this decomposition is based on an Young Itô-Kunita formula for $\alpha$-H{ö}lder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022).

15 pages