Collected mathematical papers by Cayley A.

By Cayley A.

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3) implies that for any e > 0 Stl B0 Ì O e ( A l ) l ÎL for all h -1 ( ln 1 ¤ e t* ( e , when t ³ + ln C0 ) . Here O e C0 ) º A l . 7) is an e -vicinity of the set l l inf dist ( St 1 x , St 2 y ) £ x Î B0 y Î B0 l l £ sup dist ( St 1 x , St 2 x ) £ C1 e a t dist ( l1 , l 2 ) . 8) Since A l Ì B0 , we have A l = S tl A l Ì Stl B 0 . 7) gives us that A l Ì Stl B0 Ì O e ( A l ) . 9) dist ( x , A l ) £ dist ( x , z ) + dist ( z , A l ) holds. Hence, we can find that dist ( x , A l ) £ dist ( x , z ) + e for all x Î X and z Î O e ( A l ) .

It is sufficient to verify that w ( B ) uniformly attracts the absorbing set B . Assume the contrary. Then the value sup {dist ( St y , w ( B ) ) : y Î B } does not tend to zero as t ® ¥ . This means that there exist d > 0 and a sequence { tn : tn ® ¥ } such that ì ü sup ídist ( St y , w ( B ) ) : y Î B ý ³ 2 d . n î þ Therefore, there exists an element yn Î B such that dist ( Stn yn , w ( B ) ) ³ d , n = 1, 2, ¼ . 1) As before, a convergent subsequence {Stn yn } can be extracted from the sequence k k { Stn yn } .

10 Let ( X , St ) be an asymptotically smooth dynamical system. Assume that for any bounded set B Ì X the set g + B = = t ³ 0 St ( B ) is bounded. Show that the system ( X , St ) possesses a global attractor A of the form È A= È {w (B) : B Ì X , B is bounded } . e. there exists a bounded set B 0 Ì X such that dist X ( St y , B 0 ) ® 0 as t ® ¥ for every point y Î X . Prove that the global attractor A is compact. § 6 On the Structure of Global Attractor The study of the structure of global attractor of a dynamical system is an important problem from the point of view of applications.

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