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# Spaces of dynamical systems by Sergei Yu Pilyugin

By Sergei Yu Pilyugin

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T; 0/. s// D ‰. 0/ D 0. 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.

2. F; G/ ! F; G/ ! 0, then 1. ; / ! 0. 0. a//; t 2 Œa; b: ; / ! 0. 3 Metrics on the space of systems of differential equations 21 Proof. Let x be an arbitrary point of the space Rn . 9) implies that 0. L/: This proves statement (1). To estimate the value 1 . 0/ D E, where E is the unit matrix of size n n. F; G/ ! F; G/ Ä 1. t; x/). 0/j D 1. 14) where h i denotes scalar product. t /. F; G/ ! F; G/ ! 0. L/ < ı. t; x/j Ä ı; jt j Ä 1; for any x 2 Rn . Then @F . t; x// @x @F . 3 Metrics on the space of systems of differential equations 23 for any x 2 Rn .

5. If h topologically conjugates f and g, then h. g/. Proof. x0 / and an arbitrary number N . U / is a neighborhood of the point x0 . x/ 2 V . x/. g/. x// 2 U . g/; hence, h. f /. We conclude that h. f // D A similar reasoning shows that h 1 . g/. Corollary. If homeomorphisms f and g are topologically conjugate, then they are -conjugate. M /. 3 Nonwandering set Let M be a smooth closed manifold. M / is called stable if there exists a neighborhood W of the diffeomorphism f in the C 1 -topology such that any diffeomorphism g 2 W is -conjugate with f .