Image of a shift map along the orbits of a flow

Details Image of a shift map along the orbits of a flow Image of a shift map along the orbits of a flow

2009-02-14

arxiv.org/abs/0902.2418v2

Mathematics

Dynamical Systems

Version 2. 39 pages, 5 figures. The text of the paper is essentially rewritten. All the exposition is clarified

Scientific paper

Let (F_t) be a smooth flow on a smooth manifold M and h:M-->M be a smooth orbit preserving map. The following problem is studied: suppose that for every point z of M there exists a germ of smooth function f_z at z such that near z h(x)=F_{f_z(x)}(x). Can the functions (f_z) be glued together to give a smooth function on all of M? This question is closely related to reparametrizations of flows. We describe a large class of flows for which the above problem can be resolved, and show that they have the following property: any smooth flow (G_t) whose orbits coincides with the ones of (F_t) is obtained from (F_t) by smooth reparametrization of time. The proof of our principal statement uses results of D.Hoffman and L.N.Mann about diameters of effective actions of Lie grous of Riemannian manifolds.

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