By Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)
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Extra resources for An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory
8. is the £-neighborhood of J. If T: X + X is continuous and there is a compact set which attracts compact sets of X and J = nn K TnK, then (i) J is independent of K; (ii) J is maximal, cOmpact, invariant; (iii) J is stable and attracts compact sets of X. If, in addition, T is asymptotically smooth, then (iv) for any compact set H =X, there is a neighborhood H such that is bounded and J particular, J is uniformly asymptotically stable. 8 and, in addition, gives more information about the strong attractivity properties of the set J.
N ~ K is the same for all compact sets K which attract comn>O nr pact sets of -CO, it is nonempty, compact, connected, invariant and is the The set ~= maximal compact invariant set. Proof: F Assume the hypotheses in the statement hold and fix is point dissipative, there exists a bounded set ~ E CO, there is a to = to(~) such that By continuity, for each ~ E CO such that ~tO~ c~(B,E) for is a compact map, it follows that ~t+rO~ of N(H) c B* for to(~) ~ ~t~ C~(B,E) B* t ~ to(~)+r. = ~r~(B,E) t ~ to(~)+r.
If then fl(F) fl(F) is invariant. 1. 7. If FEEt closed and, moreover, if ~r is an RFDE on a compact manifold M,~ fI(F)!!. is one-to-one on A(F), then fI(F) is invariant. Most of the results in this section are valid in a more abstract setting. We state the results without proof, for maps, and the extension to flows is easy to accomplish. Throughout the discussion is continuous. 8. is the £-neighborhood of J. If T: X + X is continuous and there is a compact set which attracts compact sets of X and J = nn K TnK, then (i) J is independent of K; (ii) J is maximal, cOmpact, invariant; (iii) J is stable and attracts compact sets of X.