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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 $α\in (1/2, 1)$. 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 $α$-H{ö}lder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022).