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Topology of Foliations and Decomposition of Stochastic Flows of Diffeomorphisms

Abstract

Let M be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno et al. (Stoch Dyn 13(4):1350009, 2013) it is shown that, up to a stopping time \(\tau \), a stochastic flow of local diffeomorphisms \(\varphi _t\) in M can be written as a Markovian process in the subgroup of diffeomorphisms which preserve the horizontal foliation composed with a process in the subgroup of diffeomorphisms which preserve the vertical foliation. Here, we discuss topological aspects of this decomposition. The main result guarantees the global decomposition of a flow if it preserves the orientation of a transversely orientable foliation. In the last section, we present an Itô-Liouville formula for subdeterminants of linearised flows. We use this formula to obtain sufficient conditions for the existence of the decomposition for all \(t\ge 0\).

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Acknowledgments

Alison M. Melo was supported by CNPq 142084/2012-3. Leandro Morgado was supported by FAPESP 11/14797-2. Paulo R. Ruffino was partially supported by FAPESP 12/18780-0, 11/50151-0 and CNPq 477861/2013-0.

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Correspondence to Paulo R. Ruffino.

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Melo, A.M., Morgado, L. & Ruffino, P.R. Topology of Foliations and Decomposition of Stochastic Flows of Diffeomorphisms. J Dyn Diff Equat 30, 39–54 (2018). https://doi.org/10.1007/s10884-016-9553-3

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Keywords

  • Stochastic flow of diffeomorphisms
  • Decomposition of diffeomorphisms
  • Biregular foliations
  • Transversely orientable foliation

Mathematics Subject Classification

  • 60H10
  • 58J65
  • 57R30