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Geometry in the neighborhood of invariant manifolds of maps and flows in linearization. (English) Zbl 0746.58008
Pitman Research Notes in Mathematics. 233. Harlow, New York: Longman Scientific & Technical, John Wiley & Sons, Inc. 89 p. (1990).
The book consists of two parts of approximately equal lengths each written by one of the authors. In the first part homomorphisms \(P: \mathbb{R}^ r\times \mathbb{R}^ t\to\mathbb{R}^ r\times\mathbb{R}^ t\) given by \(P(z,y)=(f(z,y),L(y)+Y(z,y))\) are considered where \(L: \mathbb{R}^ t\to\mathbb{R}^ t\) is linear and expanding, and \(P\) is (by Lipschitz constraints) sufficiently close to a mapping \(P_ z\times L\) with \(P_ z: \mathbb{R}^ r\to\mathbb{R}^ r\) less expanding than \(L\). Under these assumptions a horizontal and a vertical foliation \(H\), \(V\), respectively of \(\mathbb{R}^ r\times\mathbb{R}^ t\) are constructed such that \(P\) maps leaves to leaves. There is an invariant leaf of \(H\) which is the graph of a function \(G: \mathbb{R}^ r\to \mathbb{R}^ t\). Using these foliations new coordinates \(\bar z,\;\bar y\) can be introduced. (The leaves correspond to the manifolds given by \(\bar y=\text{const.}\) or \(\bar z=\text{const.}\), respectively.) With respect to the new coordinates \(P\) splits as \[ P(\bar z,\bar y)=(f(\bar z,G(\bar z)),L(\bar y)) \] and is therefore linearized with respect to \(y\). This is applied to get a corresponding linearization for flows given by autonomous differential equations on \(\mathbb{R}^ r\times\mathbb{R}^ t\). In the second part this linearization for the solution of differential equations is constructed directly (also using a horizontal and a vertical foliation).
Reviewer: H.G.Bothe (Berlin)

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
37D99 Dynamical systems with hyperbolic behavior
37C10 Dynamics induced by flows and semiflows
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)