# Asymptotic Behavior of Dissipative Systems by Jack K. Hale

By Jack K. Hale

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**Sample text**

8 are due to Mane [1981] and are extensions of Mallet-Paret [1976] (see also Hale, Magalhaes, and Oliva [1984]). Other results on Hausdorff dimension have been given by A. Douady and J. Oesterle [1980]. 9 is based on Massatt [1983a], [1983b]. 1. Limit sets. Let X be a complete metric space, i? + = [0, oo). A family of mappings T(t): X —> X, t > 0, is said to be a Cr-semigroup, r > 0, provided that (i) T(0) = /, (ii) T{t + s) = T{t)T{s), t > 0, s > 0, (iii) T(t)x is continuous in £,x together with Prechet derivatives in x up through order r for (£, x) E i?

The basic result for the estimate of c(K) is contained in THEOREM 2 . 8 . 2 . Let X be a Banach space, U C E an open set, T: U -+ E a C1 map, and K C U a compact set such that T(K) D K. T,|j, 0 < A < 1/2, 0 < a < (1/2A) - 1, v = SMVx^Kvx{pxT2). If DXT G d{E) for all x G K, then c{K) < oo. 9 Dissipativeness in tw o spaces. In the applications, it often happens that the mapping T under consideration is defined on two Banach spaces with one compactly imbedded in the other. Under these circumstances, one sometimes can obtain much more information about the asymptotic behavior of iterates of T.

If a; is a hyperbolic fixed point of T, then there is a neighborhood V of x such that Wfoc(x,T) d ^Ws(x,T,V) d =Sf {y G W ( s , r ) : T**/ G V,n > 0}, WftcOM) = f W^(s,T, V) H f {y G ^ w ( x , T ) : T~ny G 7 , n > 0} r are C -manifolds and will be referred to as the local stable and unstable manifolds. If the maps T and DT are one-to-one on X, then Ws{x, T) and Wu{x, T) are C r -manifolds immersed in X (see the Appendix). A point x G X is a periodic point of period p of T if T p x = x, X^z / z, j = 1,2,...