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Furthermore, let the following bounds hold: t;, - tn ~ N, N > - oo. We choose the quantities Tn as so to have p(f(p, tn It is clear that Tn + Tn), M) = f1n· satisfies the inequalities N ~ Tn :::;;; 0. As a result 38 STABILITY OF INVARIANT SETS [CHAP. I of this, we can choose from the sequence Tn a converging subsequence. We shall assume that Tn is from the outset such that Tn ~to. We then have from Pn ~Po. Tn ~to that f(Pn. Tn) ~/(Po, to) = qo. It follows from this that, on the one hand, q0 eM, and on the other hand f(qo, -to) = Po, PoEM, which is impossible, for the set M is invariant.

E M such that Pn --+ p. Then also f(Pn. t) --+ f(p, t). Consequently, f(p, t) EM for t E (- oo, oo). It is clear in this case that f(p, t) E M,M. COROLLARY. The boundary of an open invariant set consists of (complete) trajectories of the dynamical system f(p, t). DEFINITION. A point q is called~~ w-limit point of the motion f(p, t) if there exists a sequence of nUmbers tn --+ oo as n oo oo. The point q is called an or:such that f(p, tn) --+ q as n limit point if there exists a sequence tn - - oo such that f(p, tn) -q as n - oo.

Let us clarify this definition by means of the example given above. 15) to be uniformly attracting, it is necessary and sufficient that j,~+T g(O) dO be uniformly bounded from below relative to toE (- oo, oo) for any finite T > 0. Let us set g(t) = -ltl. Then for to ~ 0 and T ~ 0 + t = T +to. It is clear that Jxl ~ 0 as to ~ + oo, no matter what the value of T > 0. This means that the invariant set x = 0 is not uniformly attracting. THEOREM 6. An asymptotically stable closed invariant set M of a dynamical system f(p, t), having a sufficiently small compact neighborhood, is uniformly asymptotically stable and uniformly attracting.

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A similarity problem for Toeplitz operators by Clark D.N.

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