Geometric Aspects of Young Integral: Decomposition of Flows

In this paper we study geometric aspects of dynamics generated by Young differential equations (YDE) driven by α -Hölder trajectories with α ∈ ( 1 / 2 , 1 ) . We present a number of properties and geometrical constructions in this context: Young Itô geometrical formula, horizontal lift in principal fibre bundles, parallel transport, 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 Itô-Wentzel type formula for Young integration along α -Hölder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022).