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By Sergei Yu Pilyugin

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S// D ‰. 4)). The theorem is proven. Remark. t; x/ ‰. x/; x/ D ‰. x/ ! x/ ! 0 as x ! p. Thus, any trajectory that intersects a small neighborhood of the point p intersects the surface Q as well. 1 is called the local Poincaré diffeomorphism generated by the transverse surfaces P and Q. 1) (thus, it is a closed trajectory) and the surfaces P and Q coincide (precisely this case was studied by Poincaré). t; x/; dt 0;1 in R where x 2 Rn . t; x/ for some ! > 0. t0 ; x0 /. / is a solution as well. 6) can be continued to R.

M / cannot be embedded into a flow generated by a Lipschitz continuous vector field on M . 5 Immersions and embeddings In this book, we consider two main classes of mappings studied in differential topology, immersions and embeddings. 5 Immersions and embeddings embeddings of smooth manifolds (Euclidean spaces and disks in such spaces) into Euclidean spaces. Let us recall the basic definitions. Let f be a mapping of a manifold M to a manifold N . M /. x/ D dim M for any x 2 M (let us note that the last condition implies that dim N dim M ).

Remark. Sometimes, it is possible to significantly simplify a problem by passing from a homeomorphism to a topologically conjugate homeomorphism (this idea will be applied in Section 9, where we study the Smale horseshoe). Here we consider as an example two semi-dynamical systems. Let f be a continuous mapping of a topological space M into itself. , a mapping W ZC M ! M whose properties are similar to properties (DDS1)–(DDS3) (one has to replace Z by ZC in properties (DDS2) and (DDS3)). m; x/ W m 2 ZC ºI the definition of a periodic point is literally the same as in the case of a dynamical system.

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