Decomposition of discontinuous flows of diffeomorphisms: Jumpings, geometrical and topological aspects

Let [Formula: see text] be a compact manifold equipped with a pair of complementary foliations, say horizontal [Formula: see text] and vertical [Formula: see text]. In Melo, Morgado and Ruffino [Decomposition of stochastic flows generated by Stratonovich SDEs with jumps, Discrete Contin. Dyn. Syst. Ser. B 21(9) (2016) 3209–3218] it is proved that if a semimartingale [Formula: see text] has a finite number of jumps in compact intervals then, up to a stopping time [Formula: see text], a stochastic flow of local diffeomorphisms in [Formula: see text] driven by [Formula: see text] can be decomposed into a process in the Lie group of diffeomorphisms which fix the leaves of [Formula: see text] composed with a process in the Lie group of diffeomorphisms which fix the leaves of [Formula: see text]. Dynamics at the discontinuities of [Formula: see text] here are interpreted in the Marcus sense as in Kurtz, Pardoux and Protter [Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. Henri Poincar Probab. Stat. 31(2) (1995) 351–377]. Here, we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps and show explicit equations for each component. Our technique is based in an extension of the Itô–Ventzel–Kunita formula for stochastic flows with jumps. Geometrical and other topological obstructions for the decomposition are also considered: e.g., an index of attainability is introduced to measure the complexity of the dynamics with respect to the pair of foliations.